{"id":2357,"date":"2026-06-02T21:31:36","date_gmt":"2026-06-02T13:31:36","guid":{"rendered":"http:\/\/www.preluna.xyz\/?p=2357"},"modified":"2026-06-03T17:27:08","modified_gmt":"2026-06-03T09:27:08","slug":"%e9%9b%85%e5%8f%af%e6%af%94%e5%9b%a0%e5%ad%90","status":"publish","type":"post","link":"http:\/\/www.preluna.xyz\/index.php\/2026\/06\/02\/%e9%9b%85%e5%8f%af%e6%af%94%e5%9b%a0%e5%ad%90\/preluna\/text\/","title":{"rendered":"\u96c5\u53ef\u6bd4\u56e0\u5b50"},"content":{"rendered":"\n
\"\"<\/div><\/figure>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n

\u672c\u9898\u7b54\u6848\u4e3a\uff1a<\/p>\n\n\n\n

f_Z(z) = 2(1 - z),    0 < z < 1\nf_Z(z) = 0,           \u5176\u4ed6\n<\/code><\/pre>\n\n\n\n
\u8fd9\u91cc\u7684\u5173\u952e\u4e0d\u662f\u628a X<\/code> \u548c Y<\/code> \u5f53\u6210\u72ec\u7acb\u53d8\u91cf\uff0c\u800c\u662f\u76f4\u63a5\u4f7f\u7528\u4e8c\u7ef4\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u7684\u53d8\u91cf\u53d8\u6362\u516c\u5f0f\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4e00\u3001\u9898\u76ee\u6761\u4ef6\u4e0e\u53d8\u91cf\u8303\u56f4<\/h2>\n\n\n\n
\u9898\u76ee\u7ed9\u51fa (X,Y)<\/code> \u7684\u8054\u5408\u5bc6\u5ea6\uff1a<\/p>\n\n\n\n
f(x,y) = x + y,    0 < x < 1, 0 < y < 1\n       = 0,        \u5176\u4ed6\n<\/code><\/pre>\n\n\n\n
\u8981\u6c42\uff1a<\/p>\n\n\n\n
Z = XY\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a 0 < X < 1<\/code>\uff0c0 < Y < 1<\/code>\uff0c\u6240\u4ee5\uff1a<\/p>\n\n\n\n
0 < Z = XY < 1\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\u6700\u540e\u7684\u5bc6\u5ea6\u53ea\u53ef\u80fd\u5728 0 < z < 1<\/code> \u4e0a\u975e\u96f6\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4e8c\u3001\u4f7f\u7528\u4e58\u79ef\u578b\u968f\u673a\u53d8\u91cf\u516c\u5f0f<\/h2>\n\n\n\n
\u56fe 2 \u4e2d\u7684\u516c\u5f0f\u672c\u8d28\u4e0a\u662f\uff1a<\/p>\n\n\n\n
\u82e5 Z = XY\uff0c\u4e14 (X,Y) \u7684\u8054\u5408\u5bc6\u5ea6\u4e3a \u03c6(x,y)\uff0c\u5219\n\nf_Z(z) = \u222b 1\/|x| \u00b7 \u03c6(x, z\/x) dx\n<\/code><\/pre>\n\n\n\n
\u8fd9\u91cc\u4e0d\u662f\u8981\u6c42 X,Y<\/code> \u72ec\u7acb\u3002\u53ea\u8981\u77e5\u9053\u8054\u5408\u5bc6\u5ea6 \u03c6(x,y)<\/code>\uff0c\u5c31\u53ef\u4ee5\u7528\u8fd9\u4e2a\u516c\u5f0f\u3002<\/p>\n\n\n\n
\u672c\u9898\u4e2d\uff1a<\/p>\n\n\n\n
\u03c6(x,y) = x + y\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
\u03c6(x, z\/x) = x + z\/x\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b 1\/|x| \u00b7 (x + z\/x) dx\n<\/code><\/pre>\n\n\n\n
\u4f46\u79ef\u5206\u8303\u56f4\u4e0d\u80fd\u76f4\u63a5\u5199\u6210 -\u221e<\/code> \u5230 +\u221e<\/code> \u540e\u5c31\u4e0d\u7ba1\u4e86\u3002\u5fc5\u987b\u7ed3\u5408\u539f\u6765\u7684\u533a\u57df\uff1a<\/p>\n\n\n\n
0 < x < 1\n0 < y < 1\ny = z\/x\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a 0 < z < 1<\/code>\uff0c\u4e14 x > 0<\/code>\uff0c\u6240\u4ee5\uff1a<\/p>\n\n\n\n
0 < z\/x < 1\n<\/code><\/pre>\n\n\n\n
\u8fd9\u7b49\u4ef7\u4e8e\uff1a<\/p>\n\n\n\n
x > z\n<\/code><\/pre>\n\n\n\n
\u540c\u65f6\u53c8\u6709\uff1a<\/p>\n\n\n\n
x < 1\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\u79ef\u5206\u8303\u56f4\u662f\uff1a<\/p>\n\n\n\n
z < x < 1\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b[z \u5230 1] 1\/x \u00b7 (x + z\/x) dx\n<\/code><\/pre>\n\n\n\n
\u5316\u7b80\uff1a<\/p>\n\n\n\n
1\/x \u00b7 (x + z\/x)\n= 1 + z\/x\u00b2\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\uff1a<\/p>\n\n\n\n
f_Z(z)\n= \u222b[z \u5230 1] (1 + z\/x\u00b2) dx\n= \u222b[z \u5230 1] 1 dx + \u222b[z \u5230 1] z\/x\u00b2 dx\n<\/code><\/pre>\n\n\n\n
\u5206\u522b\u8ba1\u7b97\uff1a<\/p>\n\n\n\n
\u222b[z \u5230 1] 1 dx = 1 - z\n<\/code><\/pre>\n\n\n\n
\u4ee5\u53ca\uff1a<\/p>\n\n\n\n
\u222b[z \u5230 1] z\/x\u00b2 dx\n= z\u222b[z \u5230 1] x\u207b\u00b2 dx\n= z[-1\/x][z \u5230 1]\n= z(-1 + 1\/z)\n= 1 - z\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
f_Z(z) = (1 - z) + (1 - z) = 2(1 - z)\n<\/code><\/pre>\n\n\n\n
\u6700\u7ec8\uff1a<\/p>\n\n\n\n
f_Z(z) = 2(1 - z),    0 < z < 1\nf_Z(z) = 0,           \u5176\u4ed6\n<\/code><\/pre>\n\n\n\n
\n\n\n\n
\u4e09\u3001\u56fe 2 \u516c\u5f0f\u7684\u8f6c\u5316\u539f\u7406<\/h2>\n\n\n\n
\u56fe 2 \u7684\u516c\u5f0f\u6765\u81ea\u4e8c\u7ef4\u53d8\u91cf\u53d8\u6362\u3002<\/p>\n\n\n\n
\u8981\u6c42 Z = XY<\/code> \u7684\u5bc6\u5ea6\uff0c\u901a\u5e38\u8865\u5145\u4e00\u4e2a\u8f85\u52a9\u53d8\u91cf\uff0c\u4f8b\u5982\u4ee4\uff1a<\/p>\n\n\n\n
U = X\nZ = XY\n<\/code><\/pre>\n\n\n\n
\u90a3\u4e48\uff1a<\/p>\n\n\n\n
x = u\nz = xy\n<\/code><\/pre>\n\n\n\n
\u7531 z = xy<\/code> \u5f97\uff1a<\/p>\n\n\n\n
y = z\/u\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u53cd\u53d8\u6362\u4e3a\uff1a<\/p>\n\n\n\n
x = u\ny = z\/u\n<\/code><\/pre>\n\n\n\n
\u63a5\u4e0b\u6765\u8981\u7b97\u96c5\u53ef\u6bd4\u56e0\u5b50\uff1a<\/p>\n\n\n\n
\u2202(x,y)\/\u2202(u,z)\n<\/code><\/pre>\n\n\n\n
\u4e5f\u5c31\u662f\uff1a<\/p>\n\n\n\n
x = u\ny = z\/u\n<\/code><\/pre>\n\n\n\n
\u5bf9 u,z<\/code> \u6c42\u504f\u5bfc\uff1a<\/p>\n\n\n\n
\u2202x\/\u2202u = 1\n\u2202x\/\u2202z = 0\n\n\u2202y\/\u2202u = -z\/u\u00b2\n\u2202y\/\u2202z = 1\/u\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u884c\u5217\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n
| 1        0   |\n| -z\/u\u00b2   1\/u |\n\n= 1\/u\n<\/code><\/pre>\n\n\n\n
\u53d6\u7edd\u5bf9\u503c\uff1a<\/p>\n\n\n\n
|\u2202(x,y)\/\u2202(u,z)| = 1\/|u|\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a\u8fd9\u91cc u=x<\/code>\uff0c\u6240\u4ee5\u5c31\u662f\uff1a<\/p>\n\n\n\n
1\/|x|\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\u8054\u5408\u5bc6\u5ea6\u53d8\u4e3a\uff1a<\/p>\n\n\n\n
f_{U,Z}(u,z)\n= \u03c6(u, z\/u) \u00b7 1\/|u|\n<\/code><\/pre>\n\n\n\n
\u518d\u628a\u8f85\u52a9\u53d8\u91cf U<\/code> \u79ef\u6389\uff0c\u5c31\u5f97\u5230 Z<\/code> \u7684\u8fb9\u7f18\u5bc6\u5ea6\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b \u03c6(u, z\/u) \u00b7 1\/|u| du\n<\/code><\/pre>\n\n\n\n
\u628a u<\/code> \u6539\u5199\u6210 x<\/code>\uff0c\u5c31\u662f\u56fe 2 \u7684\u516c\u5f0f\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b 1\/|x| \u00b7 \u03c6(x, z\/x) dx\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u8fd9\u4e2a\u516c\u5f0f\u7684\u672c\u8d28\u662f\uff1a<\/p>\n\n\n\n
\u5148\u505a\u4e8c\u7ef4\u53d8\u91cf\u53d8\u6362\uff0c\u518d\u5bf9\u8f85\u52a9\u53d8\u91cf\u79ef\u5206\u6d88\u53bb\u3002\n<\/code><\/pre>\n\n\n\n
\n\n\n\n
\u56db\u3001\u8fd9\u4e2a\u516c\u5f0f\u6709\u54ea\u4e9b\u5e38\u89c1\u53d8\u5f0f\uff1f<\/h2>\n\n\n\n
\u8fd9\u7c7b\u9898\u7684\u6838\u5fc3\u5957\u8def\u662f\uff1a\u8981\u6c42\u4e00\u4e2a\u65b0\u53d8\u91cf Z = h(X,Y)<\/code> \u7684\u5bc6\u5ea6\uff0c\u5c31\u8865\u4e00\u4e2a\u53d8\u91cf\uff0c\u628a\u4e8c\u7ef4\u53d8\u6362\u51d1\u6210\u4e00\u4e00\u5bf9\u5e94\uff0c\u7136\u540e\u7528\u96c5\u53ef\u6bd4\u56e0\u5b50\u3002<\/p>\n\n\n\n
1. \u548c\uff1aZ = X + Y<\/h3>\n\n\n\n
\u4ee4\uff1a<\/p>\n\n\n\n
U = X\nZ = X + Y\n<\/code><\/pre>\n\n\n\n
\u5219\uff1a<\/p>\n\n\n\n
Y = Z - X\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b \u03c6(x, z - x) dx\n<\/code><\/pre>\n\n\n\n
\u8fd9\u91cc\u96c5\u53ef\u6bd4\u56e0\u5b50\u662f 1<\/code>\u3002<\/p>\n\n\n\n
\n\n\n\n
2. \u5dee\uff1aZ = X – Y<\/h3>\n\n\n\n
\u4ee4\uff1a<\/p>\n\n\n\n
U = X\nZ = X - Y\n<\/code><\/pre>\n\n\n\n
\u5219\uff1a<\/p>\n\n\n\n
Y = X - Z\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b \u03c6(x, x - z) dx\n<\/code><\/pre>\n\n\n\n
\u4e5f\u53ef\u4ee5\u4ee4 U = Y<\/code>\uff0c\u5219\uff1a<\/p>\n\n\n\n
X = Z + Y\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b \u03c6(z + y, y) dy\n<\/code><\/pre>\n\n\n\n
\u8fd9\u4e24\u4e2a\u5199\u6cd5\u672c\u8d28\u4e00\u6837\uff0c\u53ea\u662f\u9009\u7684\u8f85\u52a9\u53d8\u91cf\u4e0d\u540c\u3002<\/p>\n\n\n\n
\n\n\n\n
3. \u79ef\uff1aZ = XY<\/h3>\n\n\n\n
\u8fd9\u5c31\u662f\u56fe 2 \u7684\u516c\u5f0f\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b 1\/|x| \u00b7 \u03c6(x, z\/x) dx\n<\/code><\/pre>\n\n\n\n
\u5982\u679c X,Y<\/code> \u72ec\u7acb\uff0c\u8054\u5408\u5bc6\u5ea6\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n\n
\u03c6(x,y) = f_X(x)f_Y(y)\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\u624d\u6709\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b 1\/|x| \u00b7 f_X(x) \u00b7 f_Y(z\/x) dx\n<\/code><\/pre>\n\n\n\n
\u4f46\u6ce8\u610f\uff0c\u672c\u9898\u6ca1\u6709\u7528\u72ec\u7acb\u6027\uff0c\u56e0\u4e3a\u9898\u76ee\u5df2\u7ecf\u7ed9\u4e86\u8054\u5408\u5bc6\u5ea6 \u03c6(x,y)=x+y<\/code>\u3002<\/p>\n\n\n\n
\n\n\n\n
4. \u5546\uff1aZ = Y\/X<\/h3>\n\n\n\n
\u4ee4\uff1a<\/p>\n\n\n\n
U = X\nZ = Y\/X\n<\/code><\/pre>\n\n\n\n
\u5219\uff1a<\/p>\n\n\n\n
Y = XZ\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b |x| \u00b7 \u03c6(x, xz) dx\n<\/code><\/pre>\n\n\n\n
\u8fd9\u91cc\u7684\u96c5\u53ef\u6bd4\u56e0\u5b50\u53d8\u6210\u4e86 |x|<\/code>\uff0c\u4e0d\u662f 1\/|x|<\/code>\u3002<\/p>\n\n\n\n
\u539f\u56e0\u662f\uff1a<\/p>\n\n\n\n
\u79ef\uff1ay = z\/x\uff0c\u6240\u4ee5 dy\/dz = 1\/x\n\u5546\uff1ay = xz\uff0c\u6240\u4ee5 dy\/dz = x\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u79ef\u548c\u5546\u7684\u7cfb\u6570\u6b63\u597d\u4e0d\u540c\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4e94\u3001\u4e3a\u4ec0\u4e48\u672c\u9898\u4e0d\u80fd\u76f4\u63a5\u5957\u72ec\u7acb\u53d8\u91cf\u4e58\u79ef\u516c\u5f0f\uff1f<\/h2>\n\n\n\n
\u672c\u9898\u7684\u8054\u5408\u5bc6\u5ea6\u662f\uff1a<\/p>\n\n\n\n
f(x,y) = x + y\n<\/code><\/pre>\n\n\n\n
\u5982\u679c X,Y<\/code> \u72ec\u7acb\uff0c\u5e94\u5f53\u6709\uff1a<\/p>\n\n\n\n
f(x,y) = f_X(x)f_Y(y)\n<\/code><\/pre>\n\n\n\n
\u4f46\u672c\u9898\u4e2d\uff1a<\/p>\n\n\n\n
f_X(x) = \u222b[0 \u5230 1] (x+y) dy = x + 1\/2\n<\/code><\/pre>\n\n\n\n
f_Y(y) = \u222b[0 \u5230 1] (x+y) dx = y + 1\/2\n<\/code><\/pre>\n\n\n\n
\u5982\u679c\u72ec\u7acb\uff0c\u5219\u5e94\u6709\uff1a<\/p>\n\n\n\n
f_X(x)f_Y(y) = (x + 1\/2)(y + 1\/2)\n<\/code><\/pre>\n\n\n\n
\u8fd9\u663e\u7136\u4e0d\u7b49\u4e8e\uff1a<\/p>\n\n\n\n
x + y\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5 X,Y<\/code> \u4e0d\u72ec\u7acb\u3002<\/p>\n\n\n\n
\u4f46\u662f\u4e0d\u72ec\u7acb\u5b8c\u5168\u4e0d\u5f71\u54cd\u672c\u9898\u8ba1\u7b97\uff0c\u56e0\u4e3a\u56fe 2 \u516c\u5f0f\u672c\u6765\u5c31\u662f\u7528\u8054\u5408\u5bc6\u5ea6 \u03c6(x,y)<\/code> \u6765\u7b97\u7684\uff0c\u4e0d\u662f\u72ec\u7acb\u516c\u5f0f\u3002<\/p>\n\n\n\n
\n\n\n\n
\u516d\u3001\u8fd9\u7c7b\u9898\u7684\u5feb\u901f\u8bc6\u522b\u65b9\u6cd5<\/h2>\n\n\n\n
\u770b\u5230\u9898\u76ee\u7ed9\u51fa (X,Y)<\/code> \u7684\u8054\u5408\u5bc6\u5ea6\uff0c\u53c8\u8981\u6c42 Z = XY<\/code>\u3001Z = X\/Y<\/code>\u3001Z = X+Y<\/code>\u3001Z = X-Y<\/code> \u8fd9\u7c7b\u53d8\u91cf\u51fd\u6570\u7684\u5bc6\u5ea6\u65f6\uff0c\u4f18\u5148\u60f3\u5230\uff1a<\/p>\n\n\n\n
\u4e8c\u7ef4\u53d8\u91cf\u53d8\u6362 + \u96c5\u53ef\u6bd4\u56e0\u5b50 + \u79ef\u6389\u8f85\u52a9\u53d8\u91cf\n<\/code><\/pre>\n\n\n\n
\u5177\u4f53\u5230\u4e58\u79ef\u578b\uff1a<\/p>\n\n\n\n
Z = XY\n<\/code><\/pre>\n\n\n\n
\u76f4\u63a5\u8bb0\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b 1\/|x| \u00b7 f(x, z\/x) dx\n<\/code><\/pre>\n\n\n\n
\u7136\u540e\u6700\u91cd\u8981\u7684\u4e00\u6b65\u662f\u627e\u79ef\u5206\u8303\u56f4\uff1a<\/p>\n\n\n\n
\u539f\u533a\u57df\u6761\u4ef6 + y = z\/x\n<\/code><\/pre>\n\n\n\n
\u672c\u9898\u5c31\u662f\uff1a<\/p>\n\n\n\n
0 < x < 1\n0 < z\/x < 1\n<\/code><\/pre>\n\n\n\n
\u63a8\u51fa\uff1a<\/p>\n\n\n\n
z < x < 1\n<\/code><\/pre>\n\n\n\n
\u8fd9\u4e00\u6b65\u662f\u8fd9\u7c7b\u9898\u6700\u5bb9\u6613\u9519\u7684\u5730\u65b9\u3002\u516c\u5f0f\u672c\u8eab\u4e0d\u96be\uff0c\u96be\u70b9\u901a\u5e38\u5728\u4e8e\u628a\u539f\u6765\u7684\u4e8c\u7ef4\u533a\u57df\u6b63\u786e\u6295\u5f71\u5230\u8f85\u52a9\u53d8\u91cf\u7684\u79ef\u5206\u533a\u95f4\u4e0a\u3002<\/p>\n\n\n\n
\u8fd9\u4e2a\u96c5\u514b\u6bd4\u56e0\u5b50\uff0c\u8fd9\u4e00\u4e32\u662f\u6307\u6309\u7167\u8fd9\u79cd\u5f0f\u5b50\u2202(x,y)\/\u2202(u,z)\u6765\u63a8\u5bfc\u7684\u8bdd\uff0c\u662f\u4e0d\u662f\u53ea\u8981\u884c\u66ff\u6362\u4e4b\u540e\uff0c\u4ed6\u8fd9\u4e2a\u6c42\u56e0\u5b50\u7684\u7ed3\u679c\u5c31\u4f1a\u4f5c\u4e3a\u66ff\u6362\u8fc7\u540e\u516c\u5f0f\u7684\u7cfb\u6570\u5bf9\u5427\uff1f<\/h2>\n\n\n\n
\u5bf9\uff0c\u53ef\u4ee5\u8fd9\u6837\u7406\u89e3\uff1a\u6309 \u2202(x,y)\/\u2202(u,z)<\/code> \u8fd9\u79cd\u53cd\u53d8\u6362\u96c5\u53ef\u6bd4\u6765\u63a8\u5bfc\u65f6\uff0c\u7b97\u51fa\u6765\u7684\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7edd\u5bf9\u503c\uff0c\u5c31\u4f1a\u4f5c\u4e3a\u53d8\u91cf\u66ff\u6362\u540e\u5bc6\u5ea6\u51fd\u6570\u524d\u9762\u7684\u201c\u7cfb\u6570\u56e0\u5b50\u201d\u3002<\/strong><\/p>\n\n\n\n
\u4f46\u8981\u6ce8\u610f\u4e09\u4e2a\u9650\u5236\uff1a<\/p>\n\n\n\n
1. \u5fc5\u987b\u53d6\u7edd\u5bf9\u503c\uff1b\n2. \u5fc5\u987b\u7528\u201c\u65e7\u53d8\u91cf\u5bf9\u65b0\u53d8\u91cf\u201d\u7684\u96c5\u53ef\u6bd4\uff1a\u2202(x,y)\/\u2202(u,z)\uff1b\n3. \u8fd8\u8981\u91cd\u65b0\u786e\u5b9a\u65b0\u53d8\u91cf\u7684\u53d6\u503c\u8303\u56f4\uff0c\u4e0d\u80fd\u53ea\u6362\u516c\u5f0f\u4e0d\u6362\u8303\u56f4\u3002\n<\/code><\/pre>\n\n\n\n
\n\n\n\n
\u66f4\u51c6\u786e\u5730\u8bf4\uff0c\u539f\u6765\u6709\u8054\u5408\u5bc6\u5ea6\uff1a<\/p>\n\n\n\n
f_{X,Y}(x,y)\n<\/code><\/pre>\n\n\n\n
\u73b0\u5728\u505a\u53d8\u91cf\u53d8\u6362\uff1a<\/p>\n\n\n\n
u = u(x,y)\nz = z(x,y)\n<\/code><\/pre>\n\n\n\n
\u5982\u679c\u80fd\u53cd\u89e3\u51fa\uff1a<\/p>\n\n\n\n
x = x(u,z)\ny = y(u,z)\n<\/code><\/pre>\n\n\n\n
\u90a3\u4e48\u65b0\u53d8\u91cf (U,Z)<\/code> \u7684\u8054\u5408\u5bc6\u5ea6\u662f\uff1a<\/p>\n\n\n\n
f_{U,Z}(u,z)\n= f_{X,Y}(x(u,z), y(u,z)) \u00b7 |\u2202(x,y)\/\u2202(u,z)|\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u4f60\u8bf4\u7684\u201c\u66ff\u6362\u8fc7\u540e\u516c\u5f0f\u7684\u7cfb\u6570\u201d\u57fa\u672c\u5c31\u662f\u8fd9\u4e2a\u610f\u601d\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4ee5\u7b2c 59 \u9898\u4e3a\u4f8b\uff0c\u8981\u7b97\uff1a<\/p>\n\n\n\n
Z = XY\n<\/code><\/pre>\n\n\n\n
\u8865\u4e00\u4e2a\u8f85\u52a9\u53d8\u91cf\uff1a<\/p>\n\n\n\n
U = X\nZ = XY\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\u53cd\u89e3\uff1a<\/p>\n\n\n\n
X = U\nY = Z\/U\n<\/code><\/pre>\n\n\n\n
\u4e5f\u5c31\u662f\uff1a<\/p>\n\n\n\n
x = u\ny = z\/u\n<\/code><\/pre>\n\n\n\n
\u63a5\u4e0b\u6765\u7b97\u96c5\u53ef\u6bd4\u56e0\u5b50\uff1a<\/p>\n\n\n\n
\u2202(x,y)\/\u2202(u,z)\n<\/code><\/pre>\n\n\n\n
\u5bf9\u5e94\u77e9\u9635\u662f\uff1a<\/p>\n\n\n\n
| \u2202x\/\u2202u    \u2202x\/\u2202z |\n| \u2202y\/\u2202u    \u2202y\/\u2202z |\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a\uff1a<\/p>\n\n\n\n
x = u\ny = z\/u\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
\u2202x\/\u2202u = 1\n\u2202x\/\u2202z = 0\n\n\u2202y\/\u2202u = -z\/u\u00b2\n\u2202y\/\u2202z = 1\/u\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\uff1a<\/p>\n\n\n\n
\u2202(x,y)\/\u2202(u,z)\n=\n| 1        0   |\n| -z\/u\u00b2   1\/u |\n= 1\/u\n<\/code><\/pre>\n\n\n\n
\u53d6\u7edd\u5bf9\u503c\uff1a<\/p>\n\n\n\n
|\u2202(x,y)\/\u2202(u,z)| = 1\/|u|\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a\u8fd9\u91cc u = x<\/code>\uff0c\u6240\u4ee5\u4e5f\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n\n
1\/|x|\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\uff1a<\/p>\n\n\n\n
f_{U,Z}(u,z)\n= f_{X,Y}(u, z\/u) \u00b7 1\/|u|\n<\/code><\/pre>\n\n\n\n
\u7136\u540e\u518d\u628a\u8f85\u52a9\u53d8\u91cf u<\/code> \u79ef\u6389\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b f_{X,Y}(u, z\/u) \u00b7 1\/|u| du\n<\/code><\/pre>\n\n\n\n
\u628a u<\/code> \u6362\u540d\u53eb x<\/code>\uff0c\u5c31\u5f97\u5230\u56fe 2 \u7684\u5f62\u5f0f\uff1a<\/p>\n\n\n\n
f_Z(z) = \u222b f_{X,Y}(x, z\/x) \u00b7 1\/|x| dx\n<\/code><\/pre>\n\n\n\n
\n\n\n\n
\u8fd9\u91cc\u7684\u672c\u8d28\u53ef\u4ee5\u7528\u4e00\u53e5\u8bdd\u6982\u62ec\uff1a<\/p>\n\n\n\n
\u6982\u7387\u4e0d\u53d8\uff0c\u4f46\u5750\u6807\u9762\u79ef\u88ab\u62c9\u4f38\u6216\u538b\u7f29\u4e86\uff0c\u6240\u4ee5\u5bc6\u5ea6\u8981\u4e58\u4e0a\u9762\u79ef\u7f29\u653e\u56e0\u5b50\u3002\n<\/code><\/pre>\n\n\n\n
\u539f\u6765\u7684\u6982\u7387\u5c0f\u5757\u5927\u7ea6\u662f\uff1a<\/p>\n\n\n\n
P \u2248 f_{X,Y}(x,y) dxdy\n<\/code><\/pre>\n\n\n\n
\u53d8\u91cf\u6362\u6210 (u,z)<\/code> \u540e\uff0c\u540c\u4e00\u4e2a\u5c0f\u5757\u9762\u79ef\u6ee1\u8db3\uff1a<\/p>\n\n\n\n
dxdy = |\u2202(x,y)\/\u2202(u,z)| dudz\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
P \u2248 f_{X,Y}(x(u,z), y(u,z)) \u00b7 |\u2202(x,y)\/\u2202(u,z)| dudz\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\u65b0\u5bc6\u5ea6\u5c31\u662f\uff1a<\/p>\n\n\n\n
\u539f\u5bc6\u5ea6\u4ee3\u5165\u53cd\u53d8\u6362 \u00d7 \u96c5\u53ef\u6bd4\u56e0\u5b50\u7edd\u5bf9\u503c\n<\/code><\/pre>\n\n\n\n
\u8fd9\u5c31\u662f\u5b83\u4e3a\u4ec0\u4e48\u4f1a\u53d8\u6210\u201c\u7cfb\u6570\u201d\u7684\u539f\u56e0\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4f46\u662f\u8981\u7279\u522b\u6ce8\u610f\uff1a\u4e0d\u80fd\u968f\u4fbf\u7b97 \u2202(u,z)\/\u2202(x,y)<\/code> \u7136\u540e\u76f4\u63a5\u4e58\u3002<\/strong><\/p>\n\n\n\n
\u5982\u679c\u7b97\u7684\u662f\uff1a<\/p>\n\n\n\n
\u2202(u,z)\/\u2202(x,y)\n<\/code><\/pre>\n\n\n\n
\u90a3\u8fd9\u662f\u201c\u65b0\u53d8\u91cf\u5bf9\u65e7\u53d8\u91cf\u201d\u7684\u96c5\u53ef\u6bd4\u3002\u5bc6\u5ea6\u53d8\u6362\u4e2d\u901a\u5e38\u8981\u7528\u7684\u662f\u5b83\u7684\u5012\u6570\uff1a<\/p>\n\n\n\n
|\u2202(x,y)\/\u2202(u,z)| = 1 \/ |\u2202(u,z)\/\u2202(x,y)|\n<\/code><\/pre>\n\n\n\n
\u5f53\u7136\u524d\u63d0\u662f\u53d8\u6362\u5c40\u90e8\u4e00\u4e00\u5bf9\u5e94\u3002<\/p>\n\n\n\n
\u6bd4\u5982\u672c\u9898\uff1a<\/p>\n\n\n\n
u = x\nz = xy\n<\/code><\/pre>\n\n\n\n
\u7b97\uff1a<\/p>\n\n\n\n
\u2202(u,z)\/\u2202(x,y)\n<\/code><\/pre>\n\n\n\n
\u77e9\u9635\u4e3a\uff1a<\/p>\n\n\n\n
| \u2202u\/\u2202x    \u2202u\/\u2202y |\n| \u2202z\/\u2202x    \u2202z\/\u2202y |\n=\n| 1    0 |\n| y    x |\n<\/code><\/pre>\n\n\n\n
\u884c\u5217\u5f0f\u662f\uff1a<\/p>\n\n\n\n
x\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u53cd\u8fc7\u6765\uff1a<\/p>\n\n\n\n
|\u2202(x,y)\/\u2202(u,z)| = 1\/|x|\n<\/code><\/pre>\n\n\n\n
\u8fd9\u548c\u524d\u9762\u7b97\u51fa\u7684 1\/|u|<\/code> \u662f\u4e00\u81f4\u7684\uff0c\u56e0\u4e3a u=x<\/code>\u3002<\/p>\n\n\n\n
\n\n\n\n
\u6240\u4ee5\u8003\u7814\u91cc\u53ef\u4ee5\u8fd9\u6837\u8bb0\uff1a<\/p>\n\n\n\n
\u5982\u679c\u7528\u53cd\u89e3\u540e\u7684\u65e7\u53d8\u91cf x=x(u,z), y=y(u,z) \u6765\u7b97\u96c5\u53ef\u6bd4\uff0c\n\u90a3\u4e48\u7b97\u51fa\u6765\u7684 |\u2202(x,y)\/\u2202(u,z)| \u76f4\u63a5\u4e58\u5230\u539f\u8054\u5408\u5bc6\u5ea6\u540e\u9762\u3002\n\n\u5982\u679c\u7528\u539f\u53d8\u6362 u=u(x,y), z=z(x,y) \u6765\u7b97\u96c5\u53ef\u6bd4\uff0c\n\u90a3\u4e48\u8981\u53d6\u5012\u6570\u518d\u4e58\u3002\n<\/code><\/pre>\n\n\n\n
\u7b2c 59 \u9898\u4e2d\u5c31\u662f\uff1a<\/p>\n\n\n\n
\u65e7\u5bc6\u5ea6\uff1af(x,y) = x + y\n\n\u53cd\u53d8\u6362\uff1a\nx = u\ny = z\/u\n\n\u4ee3\u5165\u65e7\u5bc6\u5ea6\uff1a\nf(u, z\/u) = u + z\/u\n\n\u4e58\u96c5\u53ef\u6bd4\u56e0\u5b50\uff1a\nf_{U,Z}(u,z) = (u + z\/u) \u00b7 1\/|u|\n<\/code><\/pre>\n\n\n\n
\u7136\u540e\u518d\u6839\u636e\u539f\u533a\u57df\u786e\u5b9a u<\/code> \u7684\u79ef\u5206\u8303\u56f4\u3002\u56e0\u4e3a\u539f\u6765\uff1a<\/p>\n\n\n\n
0 < x < 1\n0 < y < 1\n<\/code><\/pre>\n\n\n\n
\u4ee3\u5165\uff1a<\/p>\n\n\n\n
x = u\ny = z\/u\n<\/code><\/pre>\n\n\n\n
\u5f97\u5230\uff1a<\/p>\n\n\n\n
0 < u < 1\n0 < z\/u < 1\n<\/code><\/pre>\n\n\n\n
\u5f53 0 < z < 1<\/code> \u65f6\uff0c\u6709\uff1a<\/p>\n\n\n\n
z < u < 1\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
f_Z(z)\n= \u222b[z \u5230 1] (u + z\/u) \u00b7 1\/u du\n= \u222b[z \u5230 1] (1 + z\/u\u00b2) du\n= 2(1 - z)\n<\/code><\/pre>\n\n\n\n
\n\n\n\n
\u6700\u540e\u518d\u5f3a\u8c03\u4e00\u53e5\uff1a\u96c5\u53ef\u6bd4\u56e0\u5b50\u786e\u5b9e\u4f1a\u4f5c\u4e3a\u66ff\u6362\u540e\u7684\u7cfb\u6570\uff0c\u4f46\u5b83\u4e0d\u662f\u201c\u51ed\u7a7a\u591a\u51fa\u6765\u7684\u7cfb\u6570\u201d\uff0c\u800c\u662f\u4e8c\u7ef4\u9762\u79ef\u5143\u4ece dxdy<\/code> \u53d8\u6210 dudz<\/code> \u65f6\u7684\u9762\u79ef\u7f29\u653e\u6bd4\u4f8b\u3002<\/strong><\/p>\n\n\n\n
\u8fd9\u7c7b\u9898\u6700\u5e38\u89c1\u7684\u9519\u8bef\u4e0d\u662f\u96c5\u53ef\u6bd4\u7b97\u9519\uff0c\u800c\u662f\u53ea\u8bb0\u5f97\u4e58 1\/|x|<\/code>\uff0c\u5374\u5fd8\u4e86\u91cd\u65b0\u6839\u636e\u539f\u533a\u57df\u786e\u5b9a\u79ef\u5206\u4e0a\u4e0b\u9650\u3002\u7b2c 59 \u9898\u771f\u6b63\u7684\u5173\u952e\u5c31\u662f\u4ece\uff1a<\/p>\n\n\n\n
0 < x < 1, 0 < y < 1, y = z\/x\n<\/code><\/pre>\n\n\n\n
\u63a8\u51fa\uff1a<\/p>\n\n\n\n
z < x < 1\n<\/code><\/pre>\n\n\n\n
\u8fd9\u4e2a\u8303\u56f4\u4e00\u65e6\u786e\u5b9a\uff0c\u540e\u9762\u5c31\u662f\u673a\u68b0\u79ef\u5206\u3002<\/p>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n
\u5bf9\uff0c\u8fd9\u9053\u9898\u7684\u6838\u5fc3\u7406\u89e3\u57fa\u672c\u6b63\u786e\uff0c\u4f46\u8981\u7a0d\u5fae\u4fee\u6b63\u4e00\u4e0b\u8bf4\u6cd5\uff1a<\/p>\n\n\n\n
\u8fd9\u91cc\u4e0d\u662f\u201c\u6c42 f(x,y) \u4e0d\u80fd\u76f4\u63a5\u7528\u5bc6\u5ea6\u51fd\u6570\u201d\uff0c\u800c\u662f\u8bf4\uff1a\u8981\u6c42\u7cfb\u7edf\u5bff\u547d Z = min(X,Y) \u6216 Z = max(X,Y) \u7684\u5bc6\u5ea6\u65f6\uff0c\u4e0d\u80fd\u628a X\u3001Y \u7684\u5bc6\u5ea6\u51fd\u6570\u7b80\u5355\u4ee3\u5165 min \u6216 max \u5f97\u5230 Z \u7684\u5bc6\u5ea6\u3002<\/strong><\/p>\n\n\n\n
\u56e0\u4e3a Z = min(X,Y)<\/code> \u4e0d\u662f\u4e00\u4e00\u53d8\u6362\u3002\u5f88\u591a\u4e0d\u540c\u7684 (x,y)<\/code> \u90fd\u53ef\u80fd\u5bf9\u5e94\u540c\u4e00\u4e2a z<\/code>\uff0c\u6bd4\u5982\uff1a<\/p>\n\n\n\n
min(2,5)=2<\/code><\/p>\n\n\n\n
min(2,100)=2<\/code><\/p>\n\n\n\n
min(2,2.3)=2<\/code><\/p>\n\n\n\n
\u6240\u4ee5\u4e0d\u80fd\u50cf\u4e8c\u7ef4\u53d8\u91cf\u53d8\u6362\u90a3\u6837\uff0c\u76f4\u63a5\u7528\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u5199\u51fa\u5bc6\u5ea6\u3002\u8003\u7814\u6570\u5b66\u91cc\u9047\u5230\u8fd9\u79cd\u201c\u7531\u968f\u673a\u53d8\u91cf\u6784\u9020\u65b0\u968f\u673a\u53d8\u91cf\u201d\u7684\u9898\uff0c\u5c24\u5176\u662f min(X,Y)<\/code>\u3001max(X,Y)<\/code>\uff0c\u6700\u7a33\u7684\u65b9\u6cd5\u5c31\u662f\u5148\u6c42\u5206\u5e03\u51fd\u6570\uff0c\u518d\u6c42\u5bfc\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570\u3002<\/p>\n\n\n\n
\n\n\n\n
\u672c\u9898\u4e2d\uff0cX \u548c Y \u5206\u522b\u8868\u793a\u4e24\u4e2a\u72ec\u7acb\u5b50\u7cfb\u7edf L\u2081\u3001L\u2082 \u7684\u5bff\u547d\uff0c\u4e14<\/p>\n\n\n\n
f_X(x) = \u03b1e^(-\u03b1x), x > 0<\/code><\/p>\n\n\n\n
f_Y(y) = \u03b2e^(-\u03b2y), y > 0<\/code><\/p>\n\n\n\n
\u5b83\u4eec\u90fd\u662f\u6307\u6570\u5206\u5e03\u3002\u5148\u7531\u5bc6\u5ea6\u51fd\u6570\u5f97\u5230\u5206\u5e03\u51fd\u6570\uff1a<\/p>\n\n\n\n
F_X(x) = P(X \u2264 x) = 1 - e^(-\u03b1x), x > 0<\/code><\/p>\n\n\n\n
F_Y(y) = P(Y \u2264 y) = 1 - e^(-\u03b2y), y > 0<\/code><\/p>\n\n\n\n
\u540c\u65f6\u8fd8\u8981\u8bb0\u4f4f\u751f\u5b58\u51fd\u6570\uff1a<\/p>\n\n\n\n
P(X > x) = e^(-\u03b1x)<\/code><\/p>\n\n\n\n
P(Y > y) = e^(-\u03b2y)<\/code><\/p>\n\n\n\n
\n\n\n\n
\u5bf9\u4e8e\u4e32\u8054\u7cfb\u7edf\uff0c\u53ea\u8981\u5176\u4e2d\u4e00\u4e2a\u574f\u4e86\uff0c\u6574\u4e2a\u7cfb\u7edf\u5c31\u505c\u6b62\u5de5\u4f5c\uff0c\u6240\u4ee5\u7cfb\u7edf\u5bff\u547d\u4e3a<\/p>\n\n\n\n
Z = min(X,Y)<\/code><\/p>\n\n\n\n
\u8981\u6c42 Z<\/code> \u7684\u5206\u5e03\u51fd\u6570\uff1a<\/p>\n\n\n\n
F_Z(z) = P(Z \u2264 z)<\/code><\/p>\n\n\n\n
\u4e5f\u5c31\u662f\uff1a<\/p>\n\n\n\n
F_Z(z) = P(min(X,Y) \u2264 z)<\/code><\/p>\n\n\n\n
\u8fd9\u4e2a\u4e8b\u4ef6\u4e0d\u592a\u597d\u76f4\u63a5\u7b97\uff0c\u56e0\u4e3a\u5b83\u8868\u793a\u201cX \u2264 z \u6216 Y \u2264 z\u201d\uff0c\u6d89\u53ca\u5e76\u4e8b\u4ef6\u3002\u8003\u7814\u4e2d\u901a\u5e38\u8f6c\u5316\u4e3a\u5bf9\u7acb\u4e8b\u4ef6\uff1a<\/p>\n\n\n\n
min(X,Y) > z<\/code><\/p>\n\n\n\n
\u7b49\u4ef7\u4e8e\uff1a<\/p>\n\n\n\n
X > z \u4e14 Y > z<\/code><\/p>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
P(min(X,Y) > z) = P(X > z, Y > z)<\/code><\/p>\n\n\n\n
\u7531\u4e8e X \u548c Y \u72ec\u7acb\uff1a<\/p>\n\n\n\n
P(X > z, Y > z) = P(X > z)P(Y > z)<\/code><\/p>\n\n\n\n
\u4e8e\u662f\uff1a<\/p>\n\n\n\n
P(Z > z) = e^(-\u03b1z)e^(-\u03b2z) = e^(-(\u03b1+\u03b2)z)<\/code><\/p>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
F_Z(z) = 1 - P(Z > z)<\/code><\/p>\n\n\n\n
\u5373\uff1a<\/p>\n\n\n\n
F_Z(z) = 1 - e^(-(\u03b1+\u03b2)z), z > 0<\/code><\/p>\n\n\n\n
\u518d\u5bf9 z<\/code> \u6c42\u5bfc\uff0c\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570\uff1a<\/p>\n\n\n\n
f_Z(z) = (\u03b1 + \u03b2)e^(-(\u03b1+\u03b2)z), z > 0<\/code><\/p>\n\n\n\n
\u8fd9\u8bf4\u660e\u4e32\u8054\u7cfb\u7edf\u7684\u5bff\u547d\u4ecd\u7136\u670d\u4ece\u6307\u6570\u5206\u5e03\uff0c\u53c2\u6570\u4e3a \u03b1 + \u03b2<\/code>\u3002<\/p>\n\n\n\n
\n\n\n\n
\u8fd9\u91cc\u5bb9\u6613\u8bef\u89e3\u7684\u662f\u201c\u72ec\u7acb\u4e4b\u540e\u662f\u4e0d\u662f\u53ef\u4ee5\u76f4\u63a5\u4e58\u5bc6\u5ea6\u201d\u3002<\/p>\n\n\n\n
\u72ec\u7acb\u53ea\u80fd\u8bf4\u660e\u8054\u5408\u5bc6\u5ea6\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n\n
f_XY(x,y) = f_X(x)f_Y(y) = \u03b1\u03b2e^(-\u03b1x-\u03b2y)<\/code><\/p>\n\n\n\n
\u4f46\u8fd9\u53ea\u662f (X,Y)<\/code> \u8fd9\u4e2a\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\u7684\u8054\u5408\u5bc6\u5ea6\uff0c\u4e0d\u662f Z = min(X,Y)<\/code> \u7684\u5bc6\u5ea6\u3002<\/p>\n\n\n\n
\u5982\u679c\u8981\u901a\u8fc7\u8054\u5408\u5bc6\u5ea6\u6c42 Z<\/code> \u7684\u5206\u5e03\uff0c\u4e5f\u53ef\u4ee5\uff0c\u4f46\u4ecd\u7136\u8981\u5bf9\u533a\u57df\u79ef\u5206\uff1a<\/p>\n\n\n\n
F_Z(z) = P(min(X,Y) \u2264 z)<\/code><\/p>\n\n\n\n
\u8fd9\u4e2a\u533a\u57df\u662f\uff1a<\/p>\n\n\n\n
{(x,y): x > 0, y > 0, min(x,y) \u2264 z}<\/code><\/p>\n\n\n\n
\u5b83\u4e0d\u662f\u7b80\u5355\u7684\u4e00\u6761\u7ebf\uff0c\u800c\u662f\u4e8c\u7ef4\u5e73\u9762\u4e2d\u7684\u4e00\u5927\u7247\u533a\u57df\u3002\u6240\u4ee5\u6700\u540e\u8fd8\u662f\u8981\u8f6c\u6210\u6982\u7387\u533a\u57df\u79ef\u5206\uff0c\u901a\u5e38\u6bd4\u5206\u5e03\u51fd\u6570\u6cd5\u9ebb\u70e6\u3002<\/p>\n\n\n\n
\n\n\n\n
\u5e76\u8054\u7cfb\u7edf\u5219\u76f8\u53cd\uff0c\u4e24\u4e2a\u90e8\u4ef6\u90fd\u574f\u4e86\u7cfb\u7edf\u624d\u505c\u6b62\uff0c\u6240\u4ee5\u5bff\u547d\u4e3a<\/p>\n\n\n\n
T = max(X,Y)<\/code><\/p>\n\n\n\n
\u8fd9\u65f6\u6c42\u5206\u5e03\u51fd\u6570\u7279\u522b\u65b9\u4fbf\uff1a<\/p>\n\n\n\n
F_T(t) = P(T \u2264 t)<\/code><\/p>\n\n\n\n
\u5373\uff1a<\/p>\n\n\n\n
F_T(t) = P(max(X,Y) \u2264 t)<\/code><\/p>\n\n\n\n
\u8fd9\u7b49\u4ef7\u4e8e\uff1a<\/p>\n\n\n\n
X \u2264 t \u4e14 Y \u2264 t<\/code><\/p>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
F_T(t) = P(X \u2264 t, Y \u2264 t)<\/code><\/p>\n\n\n\n
\u56e0\u4e3a X \u548c Y \u72ec\u7acb\uff1a<\/p>\n\n\n\n
F_T(t) = P(X \u2264 t)P(Y \u2264 t)<\/code><\/p>\n\n\n\n
\u4ee3\u5165\uff1a<\/p>\n\n\n\n
F_T(t) = (1 - e^(-\u03b1t))(1 - e^(-\u03b2t)), t > 0<\/code><\/p>\n\n\n\n
\u518d\u6c42\u5bfc\uff1a<\/p>\n\n\n\n
f_T(t) = \u03b1e^(-\u03b1t)(1 - e^(-\u03b2t)) + \u03b2e^(-\u03b2t)(1 - e^(-\u03b1t)), t > 0<\/code><\/p>\n\n\n\n
\n\n\n\n
\u6240\u4ee5\u8fd9\u7c7b\u9898\u53ef\u4ee5\u603b\u7ed3\u6210\u4e24\u6761\u975e\u5e38\u5e38\u8003\u7684\u89c4\u5219\uff1a<\/p>\n\n\n\n
\u4e32\u8054\u7cfb\u7edf\uff1a<\/p>\n\n\n\n
Z = min(X,Y)<\/code><\/p>\n\n\n\n
\u4f18\u5148\u7528\u751f\u5b58\u51fd\u6570\uff1a<\/p>\n\n\n\n
P(Z > z) = P(X > z, Y > z)<\/code><\/p>\n\n\n\n
\u82e5\u72ec\u7acb\uff1a<\/p>\n\n\n\n
P(Z > z) = P(X > z)P(Y > z)<\/code><\/p>\n\n\n\n
\u5e76\u8054\u7cfb\u7edf\uff1a<\/p>\n\n\n\n
T = max(X,Y)<\/code><\/p>\n\n\n\n
\u4f18\u5148\u7528\u5206\u5e03\u51fd\u6570\uff1a<\/p>\n\n\n\n
P(T \u2264 t) = P(X \u2264 t, Y \u2264 t)<\/code><\/p>\n\n\n\n
\u82e5\u72ec\u7acb\uff1a<\/p>\n\n\n\n
P(T \u2264 t) = P(X \u2264 t)P(Y \u2264 t)<\/code><\/p>\n\n\n\n
\n\n\n\n
\u56e0\u6b64\u4f60\u7684\u7406\u89e3\u53ef\u4ee5\u6574\u7406\u6210\u4e00\u53e5\u66f4\u51c6\u786e\u7684\u8bdd\uff1a<\/p>\n\n\n\n
\u5bf9\u4e8e\u7531 X\u3001Y \u6784\u6210\u7684\u65b0\u968f\u673a\u53d8\u91cf\uff0c\u7279\u522b\u662f min(X,Y)\u3001max(X,Y) \u8fd9\u79cd\u975e\u4e00\u4e00\u53d8\u6362\uff0c\u4e0d\u80fd\u628a\u539f\u6765\u7684\u5bc6\u5ea6\u51fd\u6570\u76f4\u63a5\u201c\u4ee3\u8fdb\u53bb\u201d\u5f97\u5230\u65b0\u53d8\u91cf\u5bc6\u5ea6\u3002\u901a\u5e38\u5e94\u5148\u6839\u636e\u4e8b\u4ef6\u5173\u7cfb\u6c42\u65b0\u53d8\u91cf\u7684\u5206\u5e03\u51fd\u6570\uff0c\u6700\u540e\u5bf9\u5206\u5e03\u51fd\u6570\u6c42\u5bfc\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570\u3002<\/strong><\/p>\n\n\n\n
\u8fd9\u4e2a\u9898\u5c31\u662f\u5178\u578b\u7684\u201c\u7cfb\u7edf\u5bff\u547d + \u6307\u6570\u5206\u5e03 + \u4e32\u5e76\u8054 + \u6700\u5c0f\u503c\u6700\u5927\u503c\u5206\u5e03\u201d\u7684\u8003\u6cd5\u3002\u4e32\u8054\u770b min<\/code>\uff0c\u5e76\u8054\u770b max<\/code>\uff1bmin<\/code> \u5e38\u7528\u751f\u5b58\u51fd\u6570\uff0cmax<\/code> \u5e38\u7528\u5206\u5e03\u51fd\u6570\u3002<\/p>\n\n\n\n
\u8fd9\u79cd\u7c7b\u578b\u7684\u9898\u76ee\u4e0e\u53ef\u4ee5\u76f4\u63a5\u901a\u8fc7\u6982\u7387\u5bc6\u5ea6\uff0c\u7136\u540e\u53bb\u6c42f(x,y)\u7684\u6982\u7387\u5bc6\u5ea6\u7684\u533a\u522b\u5728\u54ea\uff1f<\/h2>\n\n\n\n
\u53ef\u4ee5\u8fd9\u6837\u533a\u5206\uff1a<\/p>\n\n\n\n
\u80fd\u201c\u76f4\u63a5\u7528\u6982\u7387\u5bc6\u5ea6\u53d8\u6362\u516c\u5f0f\u201d\u7684\u9898\uff0c\u672c\u8d28\u4e0a\u662f\u53d8\u91cf\u53d8\u6362\u9898\uff1b\u800c min(X,Y)<\/code>\u3001max(X,Y)<\/code> \u8fd9\u79cd\u9898\uff0c\u672c\u8d28\u4e0a\u662f\u4e8b\u4ef6\u5173\u7cfb\u9898\u3002<\/strong><\/p>\n\n\n\n
\u8fd9\u4e24\u7c7b\u9898\u6700\u5927\u7684\u533a\u522b\u5728\u4e8e\uff1a\u539f\u6765\u7684\u968f\u673a\u53d8\u91cf\u548c\u65b0\u968f\u673a\u53d8\u91cf\u4e4b\u95f4\uff0c\u662f\u5426\u5b58\u5728\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002<\/p>\n\n\n\n
\n\n\n\n
\u5047\u8bbe\u539f\u6765\u6709\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf (X,Y)<\/code>\uff0c\u73b0\u5728\u5b9a\u4e49\u65b0\u7684\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf\uff1a<\/p>\n\n\n\n
U = X + Y<\/code><\/p>\n\n\n\n
V = Y<\/code><\/p>\n\n\n\n
\u8fd9\u65f6\u4ece (X,Y)<\/code> \u5230 (U,V)<\/code> \u662f\u53ef\u4ee5\u53cd\u89e3\u7684\uff1a<\/p>\n\n\n\n
Y = V<\/code><\/p>\n\n\n\n
X = U - V<\/code><\/p>\n\n\n\n
\u4e5f\u5c31\u662f\u8bf4\uff0c\u7ed9\u5b9a (U,V)<\/code>\uff0c\u53ef\u4ee5\u552f\u4e00\u786e\u5b9a\u539f\u6765\u7684 (X,Y)<\/code>\u3002\u8fd9\u79cd\u60c5\u51b5\u5c31\u662f\u201c\u4e8c\u7ef4\u5230\u4e8c\u7ef4\u3001\u4e00\u4e00\u5bf9\u5e94\u201d\u7684\u53d8\u91cf\u53d8\u6362\u9898\u3002<\/p>\n\n\n\n
\u4e8e\u662f\u53ef\u4ee5\u7528\u6982\u7387\u5bc6\u5ea6\u53d8\u6362\u516c\u5f0f\uff1a<\/p>\n\n\n\n
f_UV(u,v) = f_XY(u-v, v) \u00d7 |\u2202(x,y)\/\u2202(u,v)|<\/code><\/p>\n\n\n\n
\u8fd9\u91cc\u7684\u6838\u5fc3\u662f\uff1a\u65b0\u53d8\u91cf (U,V)<\/code> \u548c\u65e7\u53d8\u91cf (X,Y)<\/code> \u7ef4\u6570\u76f8\u540c\uff0c\u5e76\u4e14\u53ef\u4ee5\u4e92\u76f8\u552f\u4e00\u786e\u5b9a\u3002\u6240\u4ee5\u5bc6\u5ea6\u53ef\u4ee5\u901a\u8fc7\u201c\u4ee3\u5165 + \u96c5\u53ef\u6bd4\u56e0\u5b50\u201d\u6765\u8f6c\u5316\u3002<\/p>\n\n\n\n
\u8fd9\u7c7b\u9898\u8003\u7814\u4e2d\u5e38\u89c1\u5f62\u5f0f\u662f\uff1a<\/p>\n\n\n\n
U = X + Y, V = X - Y<\/code><\/p>\n\n\n\n
U = X\/Y, V = Y<\/code><\/p>\n\n\n\n
U = X, V = XY<\/code><\/p>\n\n\n\n
U = X + Y, V = X<\/code><\/p>\n\n\n\n
\u53ea\u8981\u80fd\u53cd\u89e3\u51fa\uff1a<\/p>\n\n\n\n
x = x(u,v)<\/code><\/p>\n\n\n\n
y = y(u,v)<\/code><\/p>\n\n\n\n
\u5e76\u4e14\u96c5\u53ef\u6bd4\u4e0d\u4e3a 0\uff0c\u5c31\u53ef\u4ee5\u6309\u53d8\u91cf\u66ff\u6362\u516c\u5f0f\u505a\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4f46\u7cfb\u7edf\u5bff\u547d\u8fd9\u9053\u9898\u4e0d\u662f\u8fd9\u79cd\u7c7b\u578b\u3002\u5b83\u5b9a\u4e49\u7684\u662f\uff1a<\/p>\n\n\n\n
Z = min(X,Y)<\/code><\/p>\n\n\n\n
\u8fd9\u662f\u4ece\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf (X,Y)<\/code> \u53d8\u6210\u4e00\u7ef4\u968f\u673a\u53d8\u91cf Z<\/code>\u3002<\/p>\n\n\n\n
\u95ee\u9898\u5728\u4e8e\uff0c\u7ed9\u5b9a Z = z<\/code>\uff0c\u65e0\u6cd5\u552f\u4e00\u53cd\u63a8\u51fa (X,Y)<\/code>\u3002<\/p>\n\n\n\n
\u4f8b\u5982\uff1a<\/p>\n\n\n\n
min(2,3) = 2<\/code><\/p>\n\n\n\n
min(2,10) = 2<\/code><\/p>\n\n\n\n
min(2,100) = 2<\/code><\/p>\n\n\n\n
min(2,2.5) = 2<\/code><\/p>\n\n\n\n
\u6240\u4ee5 Z = 2<\/code> \u5bf9\u5e94\u7684\u4e0d\u662f\u539f\u5e73\u9762\u4e2d\u7684\u4e00\u4e2a\u70b9\uff0c\u800c\u662f\u4e00\u5927\u6279\u53ef\u80fd\u7684 (X,Y)<\/code>\u3002\u8fd9\u5c31\u662f\u201c\u591a\u5bf9\u4e00\u201d\u7684\u53d8\u6362\uff0c\u4e0d\u662f\u4e00\u4e00\u53d8\u6362\u3002<\/p>\n\n\n\n
\u56e0\u6b64\u4e0d\u80fd\u5199\u6210\uff1a<\/p>\n\n\n\n
f_Z(z) = f_XY(?, ?) \u00d7 \u96c5\u53ef\u6bd4\u56e0\u5b50<\/code><\/p>\n\n\n\n
\u56e0\u4e3a\u6839\u672c\u6ca1\u6709\u552f\u4e00\u7684 x(z), y(z)<\/code> \u53ef\u4ee5\u4ee3\u8fdb\u53bb\u3002<\/p>\n\n\n\n
\n\n\n\n
\u8fd9\u91cc\u53ef\u4ee5\u7528\u4e00\u53e5\u8bdd\u6293\u4f4f\u533a\u522b\uff1a<\/p>\n\n\n\n
\u53d8\u91cf\u53d8\u6362\u516c\u5f0f\u89e3\u51b3\u7684\u662f\u201c\u70b9\u5230\u70b9\u201d\u7684\u5bc6\u5ea6\u8f6c\u79fb\uff1b\u800c min(X,Y)<\/code>\u3001max(X,Y)<\/code> \u89e3\u51b3\u7684\u662f\u201c\u533a\u57df\u4e8b\u4ef6\u201d\u7684\u6982\u7387\u7d2f\u79ef\u3002<\/strong><\/p>\n\n\n\n
\u5bf9\u4e8e Z = min(X,Y)<\/code>\uff0c\u8981\u6c42\uff1a<\/p>\n\n\n\n
F_Z(z) = P(Z \u2264 z)<\/code><\/p>\n\n\n\n
\u4e5f\u5c31\u662f\uff1a<\/p>\n\n\n\n
P(min(X,Y) \u2264 z)<\/code><\/p>\n\n\n\n
\u8fd9\u4e2a\u4e8b\u4ef6\u5bf9\u5e94\u7684\u662f\u4e8c\u7ef4\u5e73\u9762\u4e2d\u7684\u4e00\u4e2a\u533a\u57df\uff0c\u800c\u4e0d\u662f\u67d0\u4e00\u4e2a\u70b9\u3002\u6240\u4ee5\u5e94\u8be5\u5148\u7b97\u533a\u57df\u6982\u7387\uff0c\u518d\u6c42\u5bfc\u5f97\u5230\u5bc6\u5ea6\u3002<\/p>\n\n\n\n
\u66f4\u65b9\u4fbf\u7684\u662f\u7b97\u5bf9\u7acb\u4e8b\u4ef6\uff1a<\/p>\n\n\n\n
P(Z > z) = P(min(X,Y) > z)<\/code><\/p>\n\n\n\n
\u800c\uff1a<\/p>\n\n\n\n
min(X,Y) > z<\/code><\/p>\n\n\n\n
\u7b49\u4ef7\u4e8e\uff1a<\/p>\n\n\n\n
X > z \u4e14 Y > z<\/code><\/p>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
P(Z > z) = P(X > z, Y > z)<\/code><\/p>\n\n\n\n
\u82e5 X\u3001Y \u72ec\u7acb\uff0c\u5219\uff1a<\/p>\n\n\n\n
P(Z > z) = P(X > z)P(Y > z)<\/code><\/p>\n\n\n\n
\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u8fd9\u9053\u9898\u8981\u5148\u8f6c\u6210\u5206\u5e03\u51fd\u6570\u6216\u751f\u5b58\u51fd\u6570\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4e0d\u8fc7\u8981\u6ce8\u610f\u4e00\u70b9\uff1a\u8fd9\u7c7b\u9898\u4e5f\u4e0d\u662f\u5b8c\u5168\u4e0d\u80fd\u201c\u7528\u5bc6\u5ea6\u201d\u3002\u5b83\u53ea\u662f\u4e0d\u80fd\u7528\u90a3\u79cd\u7b80\u5355\u7684\u201c\u4ee3\u5165\u5bc6\u5ea6 + \u96c5\u53ef\u6bd4\u201d\u7684\u65b9\u6cd5\u3002<\/p>\n\n\n\n
\u6bd4\u5982 Z = min(X,Y)<\/code>\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u4ece\u5bc6\u5ea6\u89d2\u5ea6\u7406\u89e3\uff1a<\/p>\n\n\n\n
Z<\/code> \u5728 z<\/code> \u9644\u8fd1\u53d6\u503c\uff0c\u610f\u5473\u7740\u4e24\u79cd\u60c5\u51b5\u4e4b\u4e00\uff1a<\/p>\n\n\n\n
\u7b2c\u4e00\u79cd\uff1aX<\/code> \u5728 z<\/code> \u9644\u8fd1\u574f\u6389\uff0c\u540c\u65f6 Y > z<\/code>\uff1b<\/p>\n\n\n\n
\u7b2c\u4e8c\u79cd\uff1aY<\/code> \u5728 z<\/code> \u9644\u8fd1\u574f\u6389\uff0c\u540c\u65f6 X > z<\/code>\u3002<\/p>\n\n\n\n
\u6240\u4ee5\u5bc6\u5ea6\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n\n
f_Z(z) = f_X(z)P(Y > z) + f_Y(z)P(X > z)<\/code><\/p>\n\n\n\n
\u4ee3\u5165\u6307\u6570\u5206\u5e03\uff1a<\/p>\n\n\n\n
f_X(z) = \u03b1e^(-\u03b1z)<\/code><\/p>\n\n\n\n
f_Y(z) = \u03b2e^(-\u03b2z)<\/code><\/p>\n\n\n\n
P(X > z) = e^(-\u03b1z)<\/code><\/p>\n\n\n\n
P(Y > z) = e^(-\u03b2z)<\/code><\/p>\n\n\n\n
\u4e8e\u662f\uff1a<\/p>\n\n\n\n
f_Z(z) = \u03b1e^(-\u03b1z)e^(-\u03b2z) + \u03b2e^(-\u03b2z)e^(-\u03b1z)<\/code><\/p>\n\n\n\n
\u6574\u7406\u5f97\uff1a<\/p>\n\n\n\n
f_Z(z) = (\u03b1 + \u03b2)e^(-(\u03b1+\u03b2)z)<\/code><\/p>\n\n\n\n
\u8fd9\u548c\u5148\u6c42\u5206\u5e03\u51fd\u6570\u518d\u6c42\u5bfc\u7684\u7ed3\u679c\u4e00\u6837\u3002<\/p>\n\n\n\n
\u4f46\u662f\u8fd9\u4e2a\u201c\u5bc6\u5ea6\u6cd5\u201d\u672c\u8d28\u4e0a\u4ecd\u7136\u662f\u5728\u5206\u6790\u4e8b\u4ef6\uff0c\u800c\u4e0d\u662f\u505a\u666e\u901a\u53d8\u91cf\u66ff\u6362\u3002<\/p>\n\n\n\n
\n\n\n\n
\u6240\u4ee5\u4e24\u7c7b\u9898\u7684\u5224\u65ad\u6807\u51c6\u53ef\u4ee5\u6574\u7406\u6210\u8fd9\u6837\uff1a<\/p>\n\n\n\n
\u7c7b\u578b<\/th>\u5178\u578b\u5f62\u5f0f<\/th>\u80fd\u4e0d\u80fd\u76f4\u63a5\u7528\u5bc6\u5ea6\u53d8\u6362\u516c\u5f0f<\/th>\u539f\u56e0<\/th><\/tr><\/thead>
\u4e00\u4e00\u53d8\u91cf\u53d8\u6362<\/td>U=X+Y, V=Y<\/code><\/td>\u53ef\u4ee5<\/td>\u65b0\u65e7\u53d8\u91cf\u7ef4\u6570\u76f8\u540c\uff0c\u80fd\u552f\u4e00\u53cd\u89e3<\/td><\/tr>
\u591a\u5bf9\u4e00\u53d8\u91cf\u53d8\u6362<\/td>Z=X+Y<\/code><\/td>\u4e0d\u80fd\u76f4\u63a5\u4ee3\u5165\uff0c\u4f46\u53ef\u7528\u5377\u79ef<\/td>\u4e8c\u7ef4\u53d8\u4e00\u7ef4\uff0c\u7ed9\u5b9a Z \u4e0d\u80fd\u552f\u4e00\u786e\u5b9a X,Y<\/td><\/tr>
\u6700\u503c\u53d8\u91cf<\/td>Z=min(X,Y)<\/code>\u3001T=max(X,Y)<\/code><\/td>\u4e0d\u80fd\u76f4\u63a5\u4ee3\u5165<\/td>\u4e00\u4e2a z \u5bf9\u5e94\u4e00\u5927\u7247\u533a\u57df<\/td><\/tr>
\u4e8b\u4ef6\u578b\u6784\u9020<\/td>Z=min(X,Y)<\/code>\u3001T=max(X,Y)<\/code><\/td>\u4f18\u5148\u7528\u5206\u5e03\u51fd\u6570<\/td>\u4e8b\u4ef6\u5173\u7cfb\u6700\u6e05\u695a<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n
\u8003\u7814\u91cc\u53ef\u4ee5\u6309\u8fd9\u4e2a\u89c4\u5219\u5feb\u901f\u5224\u65ad\uff1a<\/p>\n\n\n\n
\u5982\u679c\u9898\u76ee\u8ba9\u6c42 (U,V)<\/code> \u7684\u8054\u5408\u5bc6\u5ea6\uff0c\u5e76\u4e14 U,V<\/code> \u90fd\u7531 X,Y<\/code> \u53d8\u6765\uff0c\u4f18\u5148\u8003\u8651\u201c\u53d8\u91cf\u53d8\u6362 + \u96c5\u53ef\u6bd4\u201d\u3002<\/p>\n\n\n\n
\u5982\u679c\u9898\u76ee\u53ea\u5b9a\u4e49\u4e00\u4e2a\u65b0\u53d8\u91cf\uff0c\u6bd4\u5982\uff1a<\/p>\n\n\n\n
Z = X + Y<\/code><\/p>\n\n\n\n
Z = XY<\/code><\/p>\n\n\n\n
Z = X\/Y<\/code><\/p>\n\n\n\n
Z = min(X,Y)<\/code><\/p>\n\n\n\n
Z = max(X,Y)<\/code><\/p>\n\n\n\n
\u8fd9\u65f6\u901a\u5e38\u4e0d\u80fd\u76f4\u63a5\u4ee3\u5165\u5bc6\u5ea6\u3002\u8981\u4e48\u8865\u4e00\u4e2a\u8f85\u52a9\u53d8\u91cf\u8f6c\u6210\u4e8c\u7ef4\u53d8\u6362\uff0c\u8981\u4e48\u7528\u5206\u5e03\u51fd\u6570\u6cd5\u3002<\/p>\n\n\n\n
\u5176\u4e2d\uff1a<\/p>\n\n\n\n
Z = X + Y<\/code> \u5e38\u7528\u5377\u79ef\uff1b<\/p>\n\n\n\n
Z = XY<\/code>\u3001Z = X\/Y<\/code> \u5e38\u7528\u8f85\u52a9\u53d8\u91cf\u52a0\u96c5\u53ef\u6bd4\uff1b<\/p>\n\n\n\n
Z = min(X,Y)<\/code>\u3001Z = max(X,Y)<\/code> \u5e38\u7528\u5206\u5e03\u51fd\u6570\u3002<\/p>\n\n\n\n
\n\n\n\n
\u56e0\u6b64\u8fd9\u9053\u7cfb\u7edf\u5bff\u547d\u9898\u548c\u666e\u901a\u5bc6\u5ea6\u53d8\u6362\u9898\u7684\u672c\u8d28\u533a\u522b\u5c31\u662f\uff1a<\/p>\n\n\n\n
\u666e\u901a\u5bc6\u5ea6\u53d8\u6362\u9898\u662f\u5728\u505a\u201c\u5750\u6807\u53d8\u6362\u201d\uff1b\u7cfb\u7edf\u5bff\u547d\u9898\u662f\u5728\u505a\u201c\u4e8b\u4ef6\u8f6c\u5316\u201d\u3002<\/strong><\/p>\n\n\n\n
\u4e32\u8054\u7cfb\u7edf\u5bff\u547d min(X,Y)<\/code>\uff0c\u5173\u6ce8\u7684\u662f\u201c\u81f3\u5c11\u4e00\u4e2a\u574f\u4e86\u201d\u6216\u201c\u4e24\u4e2a\u90fd\u8fd8\u6ca1\u574f\u201d\u3002<\/p>\n\n\n\n
\u5e76\u8054\u7cfb\u7edf\u5bff\u547d max(X,Y)<\/code>\uff0c\u5173\u6ce8\u7684\u662f\u201c\u4e24\u4e2a\u90fd\u574f\u4e86\u201d\u6216\u201c\u81f3\u5c11\u4e00\u4e2a\u8fd8\u6ca1\u574f\u201d\u3002<\/p>\n\n\n\n
\u6240\u4ee5\u8fd9\u7c7b\u9898\u6700\u7a33\u7684\u5165\u53e3\u4e0d\u662f\u5bc6\u5ea6\u516c\u5f0f\uff0c\u800c\u662f\u4e8b\u4ef6\u7b49\u4ef7\u5173\u7cfb\u3002<\/p>\n\n\n\n
\u76f8\u4e92\u72ec\u7acb\u6b63\u6001\u968f\u673a\u53d8\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u4ecd\u670d\u4ece\u6b63\u6001\u5206\u5e03<\/h1>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n
63 \u9898\u9009 A\uff1a\u4e0e \u03bc \u65e0\u5173\uff0c\u4e0e \u03c3\u00b2 \u6709\u5173<\/strong>\u3002<\/p>\n\n\n\n
64 \u9898\u7b54\u6848\u662f\uff1a<\/p>\n\n\n\n
\u03bc = 1\/2\n<\/code><\/pre>\n\n\n\n
\u8fd9\u4e24\u9898\u672c\u8d28\u4e0a\u90fd\u5728\u8003\u540c\u4e00\u4e2a\u77e5\u8bc6\u70b9\uff1a\u76f8\u4e92\u72ec\u7acb\u6b63\u6001\u968f\u673a\u53d8\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u4ecd\u670d\u4ece\u6b63\u6001\u5206\u5e03<\/strong>\uff0c\u5e76\u4e14\u5747\u503c\u3001\u65b9\u5dee\u6309\u7ebf\u6027\u7ec4\u5408\u89c4\u5219\u8ba1\u7b97\u3002<\/p>\n\n\n\n
\n\n\n\n
63 \u9898\u89e3\u6790<\/h2>\n\n\n\n
\u9898\u76ee\u6761\u4ef6\u662f\uff1a<\/p>\n\n\n\n
X, Y \u76f8\u4e92\u72ec\u7acb\nX ~ N(\u03bc, \u03c3\u00b2)\nY ~ N(\u03bc, \u03c3\u00b2)\n<\/code><\/pre>\n\n\n\n
\u8981\u6c42\u5224\u65ad\uff1a<\/p>\n\n\n\n
P{|X - Y| < 1}\n<\/code><\/pre>\n\n\n\n
\u662f\u5426\u4e0e \u03bc\u3001\u03c3\u00b2 \u6709\u5173\u3002<\/p>\n\n\n\n
\u56e0\u4e3a X \u548c Y \u72ec\u7acb\uff0c\u4e14\u90fd\u662f\u6b63\u6001\u5206\u5e03\uff0c\u6240\u4ee5\u5b83\u4eec\u7684\u5dee\u4ecd\u7136\u670d\u4ece\u6b63\u6001\u5206\u5e03\u3002<\/p>\n\n\n\n
\u8bbe\uff1a<\/p>\n\n\n\n
Z = X - Y\n<\/code><\/pre>\n\n\n\n
\u5219\uff1a<\/p>\n\n\n\n
E(Z) = E(X - Y) = E(X) - E(Y) = \u03bc - \u03bc = 0\n<\/code><\/pre>\n\n\n\n
\u65b9\u5dee\u4e3a\uff1a<\/p>\n\n\n\n
D(Z) = D(X - Y)\n     = D(X) + D(Y)\n     = \u03c3\u00b2 + \u03c3\u00b2\n     = 2\u03c3\u00b2\n<\/code><\/pre>\n\n\n\n
\u8fd9\u91cc\u4e4b\u6240\u4ee5\u662f\u52a0\u53f7\uff0c\u662f\u56e0\u4e3a X \u548c Y \u76f8\u4e92\u72ec\u7acb\u3002\u51cf\u6cd5\u4e0d\u4f1a\u8ba9\u65b9\u5dee\u76f8\u51cf\uff0c\u65b9\u5dee\u8861\u91cf\u7684\u662f\u6ce2\u52a8\u7a0b\u5ea6\uff0c\u7ebf\u6027\u7ec4\u5408\u91cc\u7cfb\u6570\u8981\u5e73\u65b9\uff1a<\/p>\n\n\n\n
D(aX + bY) = a\u00b2D(X) + b\u00b2D(Y)\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
X - Y ~ N(0, 2\u03c3\u00b2)\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\uff1a<\/p>\n\n\n\n
P{|X - Y| < 1}\n= P{|Z| < 1}\n<\/code><\/pre>\n\n\n\n
\u8fd9\u4e2a\u6982\u7387\u53ea\u7531 Z \u7684\u5206\u5e03\u51b3\u5b9a\uff0c\u800c Z \u7684\u5206\u5e03\u662f\uff1a<\/p>\n\n\n\n
N(0, 2\u03c3\u00b2)\n<\/code><\/pre>\n\n\n\n
\u5176\u4e2d\u5747\u503c\u5df2\u7ecf\u53d8\u6210 0\uff0c\u4e0d\u542b \u03bc\uff1b\u4f46\u65b9\u5dee\u662f 2\u03c3\u00b2\uff0c\u4ecd\u7136\u542b \u03c3\u00b2\u3002<\/p>\n\n\n\n
\u6240\u4ee5\u5b83 \u4e0e \u03bc \u65e0\u5173\uff0c\u4e0e \u03c3\u00b2 \u6709\u5173<\/strong>\u3002<\/p>\n\n\n\n
\u7b54\u6848\uff1a<\/p>\n\n\n\n
A\n<\/code><\/pre>\n\n\n\n
\u8fd9\u9053\u9898\u5bb9\u6613\u9519\u5728\u628a X - Y<\/code> \u7684\u5747\u503c\u548c\u65b9\u5dee\u6df7\u5728\u4e00\u8d77\u3002\u5747\u503c\u786e\u5b9e\u76f8\u51cf\uff0c\u5bfc\u81f4 \u03bc \u62b5\u6d88\uff1b\u4f46\u65b9\u5dee\u4e0d\u662f\u76f8\u51cf\uff0c\u800c\u662f\u56e0\u4e3a\u72ec\u7acb\u800c\u76f8\u52a0\u3002<\/p>\n\n\n\n
\n\n\n\n
64 \u9898\u89e3\u6790<\/h2>\n\n\n\n
\u9898\u76ee\u6761\u4ef6\u662f\uff1a<\/p>\n\n\n\n
X \u4e0e Y \u76f8\u4e92\u72ec\u7acb\nX ~ N(\u03bc, 1\/2)\nY ~ N(\u03bc, 1\/2)\nP{X + Y < 1} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u6c42 \u03bc\u3002<\/p>\n\n\n\n
\u8bbe\uff1a<\/p>\n\n\n\n
S = X + Y\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a X\u3001Y \u72ec\u7acb\u4e14\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u6240\u4ee5 S \u4ecd\u7136\u670d\u4ece\u6b63\u6001\u5206\u5e03\u3002<\/p>\n\n\n\n
\u5148\u6c42\u5747\u503c\uff1a<\/p>\n\n\n\n
E(S) = E(X + Y) = E(X) + E(Y) = \u03bc + \u03bc = 2\u03bc\n<\/code><\/pre>\n\n\n\n
\u518d\u6c42\u65b9\u5dee\uff1a<\/p>\n\n\n\n
D(S) = D(X + Y)\n     = D(X) + D(Y)\n     = 1\/2 + 1\/2\n     = 1\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
X + Y ~ N(2\u03bc, 1)\n<\/code><\/pre>\n\n\n\n
\u9898\u76ee\u7ed9\u51fa\uff1a<\/p>\n\n\n\n
P{X + Y < 1} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u6b63\u6001\u5206\u5e03\u6709\u4e00\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u6027\u8d28\uff1a\u6b63\u6001\u5206\u5e03\u5173\u4e8e\u5b83\u7684\u5747\u503c\u5bf9\u79f0<\/strong>\u3002<\/p>\n\n\n\n
\u5982\u679c\uff1a<\/p>\n\n\n\n
S ~ N(a, \u03c3\u00b2)\n<\/code><\/pre>\n\n\n\n
\u90a3\u4e48\uff1a<\/p>\n\n\n\n
P{S < a} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b63\u6001\u5206\u5e03\u4e2d\uff0c\u5c0f\u4e8e\u5747\u503c\u7684\u6982\u7387\u6b63\u597d\u662f 1\/2\u3002<\/p>\n\n\n\n
\u73b0\u5728\u9898\u76ee\u8bf4\uff1a<\/p>\n\n\n\n
P{S < 1} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u800c S \u7684\u5747\u503c\u662f\uff1a<\/p>\n\n\n\n
2\u03bc\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u5fc5\u987b\u6709\uff1a<\/p>\n\n\n\n
1 = 2\u03bc\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\uff1a<\/p>\n\n\n\n
\u03bc = 1\/2\n<\/code><\/pre>\n\n\n\n
\u7b54\u6848\uff1a<\/p>\n\n\n\n
1\/2\n<\/code><\/pre>\n\n\n\n
\u8fd9\u9053\u9898\u7684\u5173\u952e\u4e0d\u662f\u67e5\u8868\uff0c\u800c\u662f\u6293\u4f4f\u6b63\u6001\u5206\u5e03\u7684\u5bf9\u79f0\u6027\u3002\u770b\u5230 P{\u67d0\u4e2a\u6b63\u6001\u53d8\u91cf < \u67d0\u4e2a\u6570} = 1\/2<\/code>\uff0c\u7b2c\u4e00\u53cd\u5e94\u5e94\u8be5\u662f\uff1a\u8fd9\u4e2a\u201c\u67d0\u4e2a\u6570\u201d\u5c31\u662f\u8be5\u6b63\u6001\u53d8\u91cf\u7684\u5747\u503c\u3002<\/p>\n\n\n\n
\n\n\n\n
\u76f8\u5173\u77e5\u8bc6\u70b9\u603b\u7ed3<\/h2>\n\n\n\n
\u8fd9\u4e24\u9898\u90fd\u5c5e\u4e8e\u8003\u7814\u6570\u5b66\u6982\u7387\u8bba\u91cc\u975e\u5e38\u5e38\u89c1\u7684\u9898\u578b\uff1a\u6b63\u6001\u968f\u673a\u53d8\u91cf\u7ebf\u6027\u7ec4\u5408\u7684\u5206\u5e03<\/strong>\u3002<\/p>\n\n\n\n
\u5982\u679c\uff1a<\/p>\n\n\n\n
X ~ N(\u03bc\u2081, \u03c3\u2081\u00b2)\nY ~ N(\u03bc\u2082, \u03c3\u2082\u00b2)\n<\/code><\/pre>\n\n\n\n
\u5e76\u4e14 X\u3001Y \u76f8\u4e92\u72ec\u7acb\uff0c\u90a3\u4e48\uff1a<\/p>\n\n\n\n
aX + bY ~ N(a\u03bc\u2081 + b\u03bc\u2082, a\u00b2\u03c3\u2081\u00b2 + b\u00b2\u03c3\u2082\u00b2)\n<\/code><\/pre>\n\n\n\n
\u7279\u522b\u5e38\u7528\u7684\u51e0\u4e2a\u5f62\u5f0f\u662f\uff1a<\/p>\n\n\n\n
X + Y ~ N(\u03bc\u2081 + \u03bc\u2082, \u03c3\u2081\u00b2 + \u03c3\u2082\u00b2)\n\nX - Y ~ N(\u03bc\u2081 - \u03bc\u2082, \u03c3\u2081\u00b2 + \u03c3\u2082\u00b2)\n<\/code><\/pre>\n\n\n\n
\u6ce8\u610f\uff0cX - Y<\/code> \u7684\u65b9\u5dee\u4ecd\u7136\u662f\uff1a<\/p>\n\n\n\n
\u03c3\u2081\u00b2 + \u03c3\u2082\u00b2\n<\/code><\/pre>\n\n\n\n
\u4e0d\u662f\uff1a<\/p>\n\n\n\n
\u03c3\u2081\u00b2 - \u03c3\u2082\u00b2\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a\uff1a<\/p>\n\n\n\n
D(X - Y) = D(X) + D(-Y)\n         = D(X) + (-1)\u00b2D(Y)\n         = D(X) + D(Y)\n<\/code><\/pre>\n\n\n\n
\u8fd9\u662f\u8fd9\u7c7b\u9898\u6700\u5bb9\u6613\u51fa\u9519\u7684\u5730\u65b9\u3002<\/p>\n\n\n\n
\u5982\u679c X\u3001Y \u4e0d\u72ec\u7acb\uff0c\u5219\u8981\u591a\u51fa\u534f\u65b9\u5dee\u9879\uff1a<\/p>\n\n\n\n
D(aX + bY)\n= a\u00b2D(X) + b\u00b2D(Y) + 2abCov(X,Y)\n<\/code><\/pre>\n\n\n\n
\u4f8b\u5982\uff1a<\/p>\n\n\n\n
D(X - Y)\n= D(X) + D(Y) - 2Cov(X,Y)\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u9898\u76ee\u4e2d\u7279\u610f\u5f3a\u8c03\u201c\u76f8\u4e92\u72ec\u7acb\u201d\uff0c\u5c31\u662f\u4e3a\u4e86\u8ba9\u534f\u65b9\u5dee\u4e3a 0\uff0c\u4ece\u800c\u76f4\u63a5\u7528\u65b9\u5dee\u76f8\u52a0\u3002<\/p>\n\n\n\n
\n\n\n\n
\u6b63\u6001\u5206\u5e03\u5bf9\u79f0\u6027\u7684\u5e38\u89c1\u7528\u6cd5<\/h2>\n\n\n\n
\u5982\u679c\uff1a<\/p>\n\n\n\n
X ~ N(\u03bc, \u03c3\u00b2)\n<\/code><\/pre>\n\n\n\n
\u90a3\u4e48\u6b63\u6001\u5206\u5e03\u5173\u4e8e \u03bc \u5bf9\u79f0\uff0c\u56e0\u6b64\uff1a<\/p>\n\n\n\n
P{X < \u03bc} = 1\/2\n\nP{X > \u03bc} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u5982\u679c\u9898\u76ee\u7ed9\u51fa\uff1a<\/p>\n\n\n\n
P{X < a} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u5e76\u4e14 X \u662f\u6b63\u6001\u5206\u5e03\uff0c\u90a3\u4e48\u901a\u5e38\u53ef\u4ee5\u76f4\u63a5\u5f97\u5230\uff1a<\/p>\n\n\n\n
a = \u03bc\n<\/code><\/pre>\n\n\n\n
64 \u9898\u6b63\u662f\u8fd9\u4e2a\u8003\u6cd5\u3002<\/p>\n\n\n\n
\u5b83\u4e0d\u662f\u8ba9\u6211\u4eec\u53bb\u67e5\u6807\u51c6\u6b63\u6001\u5206\u5e03\u8868\uff0c\u800c\u662f\u8ba9\u6211\u4eec\u8bc6\u522b\uff1a<\/p>\n\n\n\n
X + Y ~ N(2\u03bc, 1)\n<\/code><\/pre>\n\n\n\n
\u53c8\u56e0\u4e3a\uff1a<\/p>\n\n\n\n
P{X + Y < 1} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5 1 \u662f X + Y<\/code> \u7684\u5747\u503c\u3002<\/p>\n\n\n\n
\n\n\n\n
\u8fd9\u7c7b\u9898\u7684\u5feb\u901f\u8bc6\u522b\u65b9\u6cd5<\/h2>\n\n\n\n
\u9047\u5230\u4e0b\u9762\u8fd9\u4e9b\u8868\u8fbe\uff0c\u8981\u7acb\u523b\u60f3\u5230\u201c\u7ebf\u6027\u7ec4\u5408\u6b63\u6001\u5206\u5e03\u201d\uff1a<\/p>\n\n\n\n
X + Y\nX - Y\naX + bY\nP{X + Y < c}\nP{|X - Y| < c}\n<\/code><\/pre>\n\n\n\n
\u89e3\u9898\u6b65\u9aa4\u56fa\u5b9a\u4e3a\u4e09\u6b65\u3002<\/p>\n\n\n\n
\u7b2c\u4e00\u6b65\uff0c\u8bbe\u65b0\u53d8\u91cf\uff1a<\/p>\n\n\n\n
Z = aX + bY\n<\/code><\/pre>\n\n\n\n
\u7b2c\u4e8c\u6b65\uff0c\u6c42\u65b0\u53d8\u91cf\u7684\u5206\u5e03\uff1a<\/p>\n\n\n\n
E(Z) = aE(X) + bE(Y)\n\nD(Z) = a\u00b2D(X) + b\u00b2D(Y)\n<\/code><\/pre>\n\n\n\n
\u5982\u679c X\u3001Y \u72ec\u7acb\uff0c\u76f4\u63a5\u76f8\u52a0\uff1b\u5982\u679c\u4e0d\u72ec\u7acb\uff0c\u8981\u8865\u534f\u65b9\u5dee\u9879\u3002<\/p>\n\n\n\n
\u7b2c\u4e09\u6b65\uff0c\u6839\u636e\u9898\u76ee\u8981\u6c42\u6c42\u6982\u7387\u6216\u53cd\u6c42\u53c2\u6570\u3002<\/p>\n\n\n\n
63 \u9898\u662f\u5224\u65ad\u6982\u7387\u4e0e\u53c2\u6570\u7684\u5173\u7cfb\uff0c\u6240\u4ee5\u53ea\u9700\u8981\u770b X - Y<\/code> \u7684\u5206\u5e03\u91cc\u8fd8\u5269\u54ea\u4e9b\u53c2\u6570\u3002<\/p>\n\n\n\n
64 \u9898\u662f\u5229\u7528 P{\u6b63\u6001\u53d8\u91cf < \u5747\u503c} = 1\/2<\/code> \u53cd\u6c42 \u03bc\u3002<\/p>\n\n\n\n
\u76f8\u5173\u7cfb\u6570 \u03c1 = 0<\/h2>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n
\u8fd9\u9898\u9009 D\uff1a1\/2<\/strong>\u3002<\/p>\n\n\n\n
\u9898\u76ee\u662f\uff1a<\/p>\n\n\n\n
(X, Y) ~ N(0, 0; \u03c3\u00b2, \u03c3\u00b2; 0)\n\n\u6c42 P{X < 2Y}\n<\/code><\/pre>\n\n\n\n
\u8fd9\u91cc\u7684\u5173\u952e\u6709\u4e24\u4e2a\uff1a<\/p>\n\n\n\n
\u7b2c\u4e00\uff0c(X, Y)<\/code> \u662f \u4e8c\u7ef4\u6b63\u6001\u5206\u5e03<\/strong>\uff0c\u4e0d\u662f\u4e00\u7ef4\u6b63\u6001\u5206\u5e03\u3002<\/p>\n\n\n\n
\u7b2c\u4e8c\uff0c\u6700\u540e\u90a3\u4e2a 0<\/code> \u8868\u793a \u76f8\u5173\u7cfb\u6570 \u03c1 = 0<\/strong>\uff0c\u4e0d\u662f\u65b0\u7684\u968f\u673a\u53d8\u91cf\uff0c\u4e5f\u4e0d\u662f\u7b2c\u4e94\u4e2a\u201c\u5206\u5e03\u53c2\u6570\u201d\u4e4b\u5916\u7684\u4e1c\u897f\u3002<\/p>\n\n\n\n
\n\n\n\n
\u4e00\u7ef4\u6b63\u6001\u5206\u5e03\u786e\u5b9e\u901a\u5e38\u5199\u6210\uff1a<\/p>\n\n\n\n
X ~ N(\u03bc, \u03c3\u00b2)\n<\/code><\/pre>\n\n\n\n
\u5b83\u53ea\u6709\u4e24\u4e2a\u53c2\u6570\uff1a<\/p>\n\n\n\n
\u03bc\uff1a\u5747\u503c\n\u03c3\u00b2\uff1a\u65b9\u5dee\n<\/code><\/pre>\n\n\n\n
\u4f46\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03\u63cf\u8ff0\u7684\u662f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf X\u3001Y \u7684\u8054\u5408\u5206\u5e03\uff0c\u56e0\u6b64\u9664\u4e86\u5206\u522b\u7ed9\u51fa X \u548c Y \u7684\u5747\u503c\u3001\u65b9\u5dee\u4e4b\u5916\uff0c\u8fd8\u8981\u8bf4\u660e\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u5173\u5173\u7cfb\u3002<\/p>\n\n\n\n
\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03\u5e38\u5199\u6210\uff1a<\/p>\n\n\n\n
(X, Y) ~ N(\u03bc\u2081, \u03bc\u2082; \u03c3\u2081\u00b2, \u03c3\u2082\u00b2; \u03c1)\n<\/code><\/pre>\n\n\n\n
\u5176\u4e2d\uff1a<\/p>\n\n\n\n
\u03bc\u2081 = E(X)\n\u03bc\u2082 = E(Y)\n\n\u03c3\u2081\u00b2 = D(X)\n\u03c3\u2082\u00b2 = D(Y)\n\n\u03c1 = X \u4e0e Y \u7684\u76f8\u5173\u7cfb\u6570\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\u9898\u76ee\u4e2d\u7684\uff1a<\/p>\n\n\n\n
(X, Y) ~ N(0, 0; \u03c3\u00b2, \u03c3\u00b2; 0)\n<\/code><\/pre>\n\n\n\n
\u610f\u601d\u662f\uff1a<\/p>\n\n\n\n
E(X) = 0\nE(Y) = 0\n\nD(X) = \u03c3\u00b2\nD(Y) = \u03c3\u00b2\n\n\u03c1XY = 0\n<\/code><\/pre>\n\n\n\n
\u6700\u540e\u4e00\u4e2a 0<\/code> \u5c31\u662f\u76f8\u5173\u7cfb\u6570\uff1a<\/p>\n\n\n\n
\u03c1XY = 0\n<\/code><\/pre>\n\n\n\n
\u5b83\u8868\u793a X \u548c Y \u4e0d\u76f8\u5173<\/strong>\u3002\u800c\u5728\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03\u4e2d\uff0c\u82e5\u76f8\u5173\u7cfb\u6570\u4e3a 0\uff0c\u5219 X \u548c Y \u8fd8\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63a8\u51fa \u76f8\u4e92\u72ec\u7acb<\/strong>\u3002<\/p>\n\n\n\n
\u8fd9\u662f\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03\u91cc\u975e\u5e38\u91cd\u8981\u7684\u7ed3\u8bba\uff1a<\/p>\n\n\n\n
\u4e8c\u7ef4\u6b63\u6001 + \u03c1 = 0  \u21d2  X \u4e0e Y \u76f8\u4e92\u72ec\u7acb\n<\/code><\/pre>\n\n\n\n
\u4f46\u8fd9\u9053\u9898\u5176\u5b9e\u5373\u4f7f\u4e0d\u5f3a\u8c03\u72ec\u7acb\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u7528\u534f\u65b9\u5dee\u4e3a 0 \u6765\u7b97\u3002<\/p>\n\n\n\n
\n\n\n\n
\u8981\u6c42\uff1a<\/p>\n\n\n\n
P{X < 2Y}\n<\/code><\/pre>\n\n\n\n
\u5148\u628a\u4e0d\u7b49\u5f0f\u79fb\u5230\u4e00\u8fb9\uff1a<\/p>\n\n\n\n
X < 2Y\n\nX - 2Y < 0\n<\/code><\/pre>\n\n\n\n
\u4ee4\uff1a<\/p>\n\n\n\n
Z = X - 2Y\n<\/code><\/pre>\n\n\n\n
\u56e0\u4e3a X\u3001Y \u670d\u4ece\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03\uff0c\u6240\u4ee5\u5b83\u4eec\u7684\u7ebf\u6027\u7ec4\u5408\u4ecd\u7136\u670d\u4ece\u6b63\u6001\u5206\u5e03\u3002\u4e5f\u5c31\u662f\u8bf4\uff1a<\/p>\n\n\n\n
Z = X - 2Y\n<\/code><\/pre>\n\n\n\n
\u4e00\u5b9a\u8fd8\u662f\u6b63\u6001\u968f\u673a\u53d8\u91cf\u3002<\/p>\n\n\n\n
\u4e0b\u9762\u6c42 Z \u7684\u5747\u503c\u548c\u65b9\u5dee\u3002<\/p>\n\n\n\n
\u5747\u503c\uff1a<\/p>\n\n\n\n
E(Z) = E(X - 2Y)\n     = E(X) - 2E(Y)\n     = 0 - 2 \u00d7 0\n     = 0\n<\/code><\/pre>\n\n\n\n
\u65b9\u5dee\uff1a<\/p>\n\n\n\n
D(Z) = D(X - 2Y)\n     = D(X) + 4D(Y) - 4Cov(X, Y)\n<\/code><\/pre>\n\n\n\n
\u53c8\u56e0\u4e3a\u76f8\u5173\u7cfb\u6570\uff1a<\/p>\n\n\n\n
\u03c1XY = 0\n<\/code><\/pre>\n\n\n\n
\u800c\uff1a<\/p>\n\n\n\n
Cov(X, Y) = \u03c1XY \u00b7 \u03c3X \u00b7 \u03c3Y\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
Cov(X, Y) = 0\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\uff1a<\/p>\n\n\n\n
D(Z) = \u03c3\u00b2 + 4\u03c3\u00b2\n     = 5\u03c3\u00b2\n<\/code><\/pre>\n\n\n\n
\u6240\u4ee5\uff1a<\/p>\n\n\n\n
Z = X - 2Y ~ N(0, 5\u03c3\u00b2)\n<\/code><\/pre>\n\n\n\n
\u4e8e\u662f\u539f\u6982\u7387\u53d8\u6210\uff1a<\/p>\n\n\n\n
P{X < 2Y}\n= P{X - 2Y < 0}\n= P{Z < 0}\n<\/code><\/pre>\n\n\n\n
\u800c Z \u662f\u5747\u503c\u4e3a 0 \u7684\u6b63\u6001\u5206\u5e03\u3002\u6b63\u6001\u5206\u5e03\u5173\u4e8e\u5747\u503c\u5bf9\u79f0\uff0c\u6240\u4ee5\uff1a<\/p>\n\n\n\n
P{Z < 0} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u56e0\u6b64\uff1a<\/p>\n\n\n\n
P{X < 2Y} = 1\/2\n<\/code><\/pre>\n\n\n\n
\u9009\uff1a<\/p>\n\n\n\n
D\n<\/code><\/pre>\n\n\n\n
\n\n\n\n
\u8fd9\u9053\u9898\u6700\u5bb9\u6613\u9519\u7684\u5730\u65b9\u5728\u4e8e\u770b\u5230 X < 2Y<\/code> \u540e\uff0c\u60f3\u628a\u5b83\u53d8\u6210\uff1a<\/p>\n\n\n\n
P{X < 2Y} = F_X(2Y)\n<\/code><\/pre>\n\n\n\n
\u8fd9\u662f\u4e0d\u5bf9\u7684\u3002\u56e0\u4e3a 2Y<\/code> \u4e0d\u662f\u4e00\u4e2a\u5e38\u6570\uff0c\u800c\u662f\u968f\u673a\u53d8\u91cf\u3002\u4e0d\u80fd\u76f4\u63a5\u628a\u968f\u673a\u53d8\u91cf\u4ee3\u5165 X \u7684\u5206\u5e03\u51fd\u6570\u3002<\/p>\n\n\n\n
\u8fd9\u7c7b\u9898\u7684\u6807\u51c6\u505a\u6cd5\u662f\uff1a\u770b\u5230\u6b63\u6001\u53d8\u91cf\u4e4b\u95f4\u7684\u4e0d\u7b49\u5f0f\uff0c\u4f18\u5148\u628a\u5b83\u6574\u7406\u6210\u4e00\u4e2a\u7ebf\u6027\u7ec4\u5408\u3002<\/p>\n\n\n\n
\u4f8b\u5982\uff1a<\/p>\n\n\n\n
P{X < 2Y}\n<\/code><\/pre>\n\n\n\n
\u8981\u8f6c\u5316\u4e3a\uff1a<\/p>\n\n\n\n
P{X - 2Y < 0}\n<\/code><\/pre>\n\n\n\n
\u7136\u540e\u7814\u7a76\uff1a<\/p>\n\n\n\n
Z = X - 2Y\n<\/code><\/pre>\n\n\n\n
\u7684\u5206\u5e03\u3002<\/p>\n\n\n\n
\u53ea\u8981 (X, Y)<\/code> \u662f\u4e8c\u7ef4\u6b63\u6001\uff0c\u90a3\u4e48 aX + bY<\/code> \u4e00\u5b9a\u662f\u4e00\u7ef4\u6b63\u6001\u3002\u63a5\u4e0b\u6765\u53ea\u9700\u8981\u6c42\u5747\u503c\u548c\u65b9\u5dee\uff0c\u518d\u7528\u6b63\u6001\u5206\u5e03\u6027\u8d28\u5373\u53ef\u3002<\/p>\n\n\n\n
\n\n\n\n
\u5173\u4e8e\u4e8c\u7ef4\u6b63\u6001\u7684\u4e94\u4e2a\u53c2\u6570\uff0c\u53ef\u4ee5\u8fd9\u6837\u8bb0\uff1a<\/p>\n\n\n\n
\u4e24\u4e2a\u5747\u503c + \u4e24\u4e2a\u65b9\u5dee + \u4e00\u4e2a\u76f8\u5173\u7cfb\u6570\n<\/code><\/pre>\n\n\n\n
\u5373\uff1a<\/p>\n\n\n\n
N(\u03bc\u2081, \u03bc\u2082; \u03c3\u2081\u00b2, \u03c3\u2082\u00b2; \u03c1)\n<\/code><\/pre>\n\n\n\n
\u5176\u4e2d \u03c1 \u7684\u4f5c\u7528\u662f\u523b\u753b X \u548c Y \u4e4b\u95f4\u7684\u7ebf\u6027\u76f8\u5173\u7a0b\u5ea6\u3002<\/p>\n\n\n\n
\u5982\u679c\uff1a<\/p>\n\n\n\n
\u03c1 > 0\n<\/code><\/pre>\n\n\n\n
\u8bf4\u660e X \u548c Y \u6b63\u76f8\u5173\u3002<\/p>\n\n\n\n
\u5982\u679c\uff1a<\/p>\n\n\n\n
\u03c1 < 0\n<\/code><\/pre>\n\n\n\n
\u8bf4\u660e X \u548c Y \u8d1f\u76f8\u5173\u3002<\/p>\n\n\n\n
\u5982\u679c\uff1a<\/p>\n\n\n\n
\u03c1 = 0\n<\/code><\/pre>\n\n\n\n
\u8bf4\u660e X \u548c Y \u4e0d\u76f8\u5173\u3002<\/p>\n\n\n\n
\u5bf9\u4e8e\u4e00\u822c\u968f\u673a\u53d8\u91cf\uff0c\u201c\u4e0d\u76f8\u5173\u201d\u4e0d\u4e00\u5b9a\u7b49\u4e8e\u201c\u72ec\u7acb\u201d\uff1b\u4f46\u5bf9\u4e8e\u4e8c\u7ef4\u6b63\u6001\u5206\u5e03\u6765\u8bf4\uff1a<\/p>\n\n\n\n
\u03c1 = 0 \u21d4 X \u4e0e Y \u76f8\u4e92\u72ec\u7acb\n<\/code><\/pre>\n\n\n\n
\u8fd9\u662f\u8003\u7814\u6982\u7387\u8bba\u91cc\u5f88\u5e38\u8003\u7684\u7ed3\u8bba\u3002<\/p>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n
\u7b2c79\u9898\u4e0d\u9009 A\uff0c\u6838\u5fc3\u539f\u56e0\u662f\uff1aY = 1\/3<\/code> \u5bf9\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u6765\u8bf4\u662f\u4e00\u4e2a\u6982\u7387\u4e3a 0 \u7684\u4e8b\u4ef6\uff0c\u4e0d\u80fd\u76f4\u63a5\u5957\u7528\u666e\u901a\u6761\u4ef6\u6982\u7387\u516c\u5f0f\u3002<\/p>\n\n\n\n
\u666e\u901a\u6761\u4ef6\u6982\u7387\u516c\u5f0f\u662f\uff1a<\/p>\n\n\n\n
P(A | B) = P(AB) \/ P(B)<\/code><\/p>\n\n\n\n
\u4f46\u8fd9\u4e2a\u516c\u5f0f\u8981\u6c42\uff1a<\/p>\n\n\n\n
P(B) > 0<\/code><\/p>\n\n\n\n
\u672c\u9898\u4e2d Y<\/code> \u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u6240\u4ee5\u53d6\u67d0\u4e00\u4e2a\u5177\u4f53\u503c\u7684\u6982\u7387\u4e3a 0\uff1a<\/p>\n\n\n\n
P(Y = 1\/3) = 0<\/code><\/p>\n\n\n\n
\u540c\u65f6\uff1a<\/p>\n\n\n\n
P(X < 1\/2, Y = 1\/3) = 0<\/code><\/p>\n\n\n\n
\u6240\u4ee5 A \u9009\u9879\u5b9e\u9645\u4e0a\u53d8\u6210\uff1a<\/p>\n\n\n\n
0 \/ 0<\/code><\/p>\n\n\n\n
\u8fd9\u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u4e0d\u80fd\u4f5c\u4e3a\u6761\u4ef6\u6982\u7387\u7684\u5b9a\u4e49\u3002<\/p>\n\n\n\n
\n\n\n\n
\u8fd9\u7c7b\u9898\u8003\u7684\u662f\u201c\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u5728\u7ed9\u5b9a Y = y \u6761\u4ef6\u4e0b\u7684\u6761\u4ef6\u5206\u5e03\u201d\u3002\u867d\u7136 P(Y = 1\/3) = 0<\/code>\uff0c\u4f46\u53ef\u4ee5\u901a\u8fc7\u6761\u4ef6\u5bc6\u5ea6\u6765\u5b9a\u4e49\uff1a<\/p>\n\n\n\n
f<sub>X|Y<\/sub>(x | y) = f(x,y) \/ f<sub>Y<\/sub>(y)<\/code><\/p>\n\n\n\n
\u5176\u4e2d\uff1a<\/p>\n\n\n\n
f<sub>Y<\/sub>(y) = \u222b<sub>-\u221e<\/sub><sup>+\u221e<\/sup> f(x,y) dx<\/code><\/p>\n\n\n\n
\u56e0\u6b64\uff1a<\/p>\n\n\n\n
P(X < 1\/2 | Y = 1\/3)<\/code><\/p>\n\n\n\n
\u5e94\u8be5\u5199\u6210\uff1a<\/p>\n\n\n\n
\u222b<sub>-\u221e<\/sub><sup>1\/2<\/sup> f<sub>X|Y<\/sub>(x | 1\/3) dx<\/code><\/p>\n\n\n\n
\u6240\u4ee5\u9009 B<\/strong>\u3002<\/p>\n\n\n\n
\n\n\n\n
A \u9009\u9879\u7684\u9519\u8bef\u672c\u8d28\u662f\uff1a\u628a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u601d\u8def\u76f4\u63a5\u5957\u5230\u4e86\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u4e0a\u3002<\/p>\n\n\n\n
\u5982\u679c Y<\/code> \u662f\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5e76\u4e14 P(Y = 1\/3) > 0<\/code>\uff0c\u90a3\u4e48 A \u662f\u53ef\u4ee5\u7528\u7684\uff1a<\/p>\n\n\n\n
P(X < 1\/2 | Y = 1\/3) = P(X < 1\/2, Y = 1\/3) \/ P(Y = 1\/3)<\/code><\/p>\n\n\n\n
\u4f46\u672c\u9898\u660e\u786e\u8bf4 (X,Y)<\/code> \u662f\u4e8c\u7ef4\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u6240\u4ee5 Y = 1\/3<\/code> \u8fd9\u79cd\u201c\u7cbe\u786e\u53d6\u503c\u4e8b\u4ef6\u201d\u6982\u7387\u4e3a 0\uff0cA \u4e0d\u6210\u7acb\u3002<\/p>\n\n\n\n
\n\n\n\n
\u987a\u5e26\u770b\u4e00\u4e0b C \u4e3a\u4ec0\u4e48\u4e5f\u4e0d\u5bf9\u3002<\/p>\n\n\n\n
C \u662f\uff1a<\/p>\n\n\n\n
\u222b<sub>-\u221e<\/sub><sup>1\/2<\/sup> f(x,y) dx<\/code><\/p>\n\n\n\n
\u5982\u679c\u8fd9\u91cc\u628a y<\/code> \u7406\u89e3\u6210 1\/3<\/code>\uff0c\u90a3\u5c31\u662f\uff1a<\/p>\n\n\n\n
\u222b<sub>-\u221e<\/sub><sup>1\/2<\/sup> f(x,1\/3) dx<\/code><\/p>\n\n\n\n
\u5b83\u79ef\u5206\u7684\u662f\u8054\u5408\u5bc6\u5ea6\u5728\u76f4\u7ebf y = 1\/3<\/code> \u4e0a\u7684\u622a\u9762\uff0c\u4e0d\u662f\u6982\u7387\u3002\u56e0\u4e3a\u4e8c\u7ef4\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u6c42\u6982\u7387\u5fc5\u987b\u5bf9\u4e00\u4e2a\u201c\u9762\u79ef\u533a\u57df\u201d\u79ef\u5206\uff0c\u800c\u4e0d\u662f\u53ea\u6cbf\u7740\u4e00\u6761\u7ebf\u79ef\u5206\u3002<\/p>\n\n\n\n
\u5b83\u548c\u6b63\u786e\u7b54\u6848\u7684\u5173\u7cfb\u662f\uff1a<\/p>\n\n\n\n
\u222b<sub>-\u221e<\/sub><sup>1\/2<\/sup> f(x,1\/3) dx = f<sub>Y<\/sub>(1\/3) \u00b7 P(X < 1\/2 | Y = 1\/3)<\/code><\/p>\n\n\n\n
\u6240\u4ee5\u5b83\u5c11\u9664\u4ee5\u4e86\u4e00\u4e2a f<sub>Y<\/sub>(1\/3)<\/code>\u3002<\/p>\n\n\n\n
\n\n\n\n
\u8fd9\u7c7b\u9898\u7684\u5feb\u901f\u5224\u65ad\u65b9\u6cd5\u662f\uff1a<\/p>\n\n\n\n
\u770b\u5230\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u91cc\u51fa\u73b0\uff1a<\/p>\n\n\n\n
P(X < a | Y = b)<\/code><\/p>\n\n\n\n
\u4e0d\u8981\u7528\uff1a<\/p>\n\n\n\n
P(X < a, Y = b) \/ P(Y = b)<\/code><\/p>\n\n\n\n
\u800c\u8981\u7acb\u523b\u60f3\u5230\u6761\u4ef6\u5bc6\u5ea6\uff1a<\/p>\n\n\n\n
P(X < a | Y = b) = \u222b<sub>-\u221e<\/sub><sup>a<\/sup> f<sub>X|Y<\/sub>(x | b) dx<\/code><\/p>\n\n\n\n
\u5176\u4e2d\uff1a<\/p>\n\n\n\n
f<sub>X|Y<\/sub>(x | b) = f(x,b) \/ f<sub>Y<\/sub>(b)<\/code><\/p>\n\n\n\n
\u8fd9\u5c31\u662f\u672c\u9898\u4e0d\u9009 A\uff0c\u800c\u9009 B \u7684\u6839\u672c\u539f\u56e0\u3002<\/p>\n\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
\u672c\u9898\u7b54\u6848\u4e3a\uff1a \u8fd9\u91cc\u7684\u5173\u952e\u4e0d\u662f\u628a X \u548c Y \u5f53\u6210\u72ec\u7acb\u53d8\u91cf\uff0c\u800c\u662f\u76f4\u63a5\u4f7f\u7528\u4e8c\u7ef4\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u7684\u53d8\u91cf\u53d8\u6362\u516c\u5f0f\u3002 \u4e00\u3001 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[54],"tags":[],"class_list":["post-2357","post","type-post","status-publish","format-standard","hentry","category-text"],"_links":{"self":[{"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2357","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=2357"}],"version-history":[{"count":4,"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2357\/revisions"}],"predecessor-version":[{"id":2370,"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2357\/revisions\/2370"}],"wp:attachment":[{"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=2357"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=2357"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.preluna.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=2357"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}