设二维随机变量（X,Y）的概率密度为f（x，y）=2-x-y 设二维随机变量(X,Y)的概率密度为:f(x,y)=4.8y...
\u8bbe\u4e8c\u7ef4\u968f\u673a\u53d8\u91cf x y \u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3afxy=2-x-y\uff0c\u89e3\uff1a(1)\u4f9d\u636e\u5b9a\u4e49\uff0cX\u7684\u8fb9\u7f18\u5bc6\u5ea6fX(x)=\u222b(-\u221e,\u221e)f(x,y)dy=\u222b(0,1)(2-x-y)dy=3/2-x\u3002\u540c\u7406\uff0cX\u7684\u8fb9\u7f18\u5bc6\u5ea6fY(y)=\u222b(-\u221e,\u221e)f(x,y)dx=\u222b(0,1)(2-x-y)dx=3/2-y\u3002
\u663e\u7136\uff0cfX(x)*fY(y)\u2260f(x,y)\uff0c\u2234X\u3001Y\u4e0d\u76f8\u4e92\u72ec\u7acb\u3002
(2)\uff0cP(X+Y\u22641)=P(X\u22641-Y)=\u222b(0,1)dy\u222b(0,1-y)(2-x-y)dx=(1/2)\u222b(0,1)(3-4y+y²)dy=2/3\u3002
\u4f9b\u53c2\u8003\u3002
\u89e3\uff1a
f(y)=
\u222b(-\u221e\u5230\u221e)f(x,y)dx
=\u222b(y\u52301)4.8y(2-x)dx
=2.4xy(4-x)|(y\u52301)
=2.4y(3-4y+y²) (0
\u5173\u4e8ex\u7684\u8fb9\u9645\u5bc6\u5ea6\u51fd\u6570Px(x):
\u5f530\u2264x\u22641\u65f6
Px(x)=\u222bf(x,y)dy,\u5173\u4e8ey\u4ece\uff0d\u221e\u79ef\u5230\uff0b\u221e=\u222b\uff082-x-y\uff09dy,\u5173\u4e8ey\u4ece0\u79ef\u52301
\u5176\u4e2d\u539f\u51fd\u6570\u4e3a\uff1a\uff082*y-x*y-y²/2\uff09
Px(x)=(2-x-½)-0=3/2-x
\u5f53x\uff1e1\u6216\u8005x\uff1c0\u65f6
Px(x)=0
\u5173\u4e8ey\u7684\u8fb9\u9645\u5bc6\u5ea6\u51fd\u6570Py(y):
\u5f530\u2264x\u22641\u65f6
Py(y)=\u222bf(x,y)dx,\u5173\u4e8ex\u4ece\uff0d\u221e\u79ef\u5230\uff0b\u221e=\u222b\uff082-x-y\uff09x,\u5173\u4e8ex\u4ece0\u79ef\u52301
\u5176\u4e2d\u539f\u51fd\u6570\u4e3a\uff1a\uff082*x-x²/2-x*y\uff09
Py(y)=(2-½-y)-0=3/2-y
\u5f53y\uff1e1\u6216\u8005y\uff1c0\u65f6
Py(y)=0
\u6269\u5c55\u8d44\u6599
\u6c42\u8fb9\u7f18\u6982\u7387\u5bc6\u5ea6\u7684\u65b9\u6cd5\uff1a
\u6c42y\u7684\u8fb9\u7f18\u5bc6\u5ea6,\u5bf9x\u4f5c\u5168\u79ef\u5206\uff0c\u6c42x\u7684\u8fb9\u7f18\u5bc6\u5ea6,\u5bf9y\u4f5c\u5168\u79ef\u5206\uff0c\u5168\u90e8\u662f\u5e38\u6570\u8303\u56f4\u5f88\u5bb9\u6613\u5224\u65ad\uff0c\u5982\u679c\u6709\u975e\u77e9\u5f62\u8303\u56f4\u7684\u8054\u5408\u5bc6\u5ea6\u51fd\u6570\u3002
\u4f8b\uff1a
\u6982\u7387\u8f6c\u5316\u4e3a\u9762\u79ef\uff1a
\u8054\u5408\u6982\u7387P(X=a,Y=b)\uff0c\u6ee1\u8db3X=a\u4e14Y=b\u7684\u9762\u79ef\uff0c\u8fb9\u7f18\u6982\u7387P(X=a)\uff0c\u4e0d\u8003\u8651Y\u7684\u53d6\u503c\uff0c\u6240\u6709\u6ee1\u8db3X=a\u7684\u533a\u57df\u7684\u603b\u9762\u79ef\uff0c\u6761\u4ef6\u6982\u7387P(X=a|Y=b)\uff0c\u5728Y=b\u7684\u524d\u63d0\u4e0b\uff0c\u6ee1\u8db3X=a\u7684\u9762\u79ef\uff08\u6bd4\u4f8b\uff09\u3002
=∫(0~1){ (2-y)x-x²/2 |(2y~1) } dy
=∫(0~1)(2-y)(1-2y)-(1-4y²)/2 dy
=∫(0~1) 2-5y+2y²-(1-4y²)/2 dy
= 2y-2.5y²+(2/3)y³-(y-4y³/3)/2 (y=1)
=2-2.5+2/3-(-1/6)
=5/6-0.5
=4/12
=1/3
z=x+y
当z=x+y<=1时
x不超过z
卷积
∫(0~z)f(x,z-x) dx
=∫(0~z) 2-z dx
= (2-z)x (x:0~z)
=(2-z)z
当z=x+y>1时
卷积
x~(z-1~1) 就是方块右上角
∫(z-1~1) f(x,z-x) dx
= (2-z)(2-z+1)
=(2-z)²
fz(z)=(2-z)z (0<x<=1)
=(2-z)² (1<x<2)
=0 else
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绛旓細P(X=0)=1/2+b,P(X+Y=1)=a+b,P(X=0,X+Y=1)=b 鈭祘X=0}涓巤X+Y=1}鐩镐簰鐙珛 鈭碢(X=0)路P(X+Y=1)=P(X=0,X+Y=1)鈭(1/2+b)(a+b)=b 鍙堚埖 1/2+1/4+a+b=1 鎵浠ワ細a=1/12 b=1/6
绛旓細EX=鈭埆[0<=y<=x<=1] xf(x,y)dxdy=鈭玔0->1]鈭玔0->x] 12xy²dydx=4/5 EY=鈭埆[0<=y<=x<=1] yf(x,y)dxdy=鈭玔0->1]鈭玔0->x] 12y³dydx=3/5 E(X²+Y²)=鈭埆[0<=y<=x<=1] (x²+y²)f(x,y)dxdy=鈭玔0->1]鈭玔0->x...
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绛旓細濡傚彸鍥炬墍绀猴細灏姒傜巼杞寲涓洪潰绉疍涓婄Н鍒嗭紝鐢遍璁撅紝鏈夛細P{X+Y鈮1}=?Df(x锛寉)dxdy鈺愨埆120(6x?12x2)dx锛14锛屾晠绛旀涓猴細14锛
绛旓細0,1)dx鈭(0,x)(x+y)f(x,y)dy=鈭(0,1)3x²dx=1銆(2)锛孍(XY)=鈭(0,1)dx鈭(0,x)xyf(x,y)dy=鈭(0,1)x³dx=1/4銆(3)锛屸埖x+y=1涓巠=x鐨勪氦鐐逛负(1/2,1/2)銆侾(X+Y鈮1)=鈭(0,1/2)dy鈭(y,1-y)f(x,y)dy=2鈭(0,1/2)锛1-2y)dy=1/2銆
绛旓細绠鍗曡绠椾竴涓嬪嵆鍙紝绛旀濡傚浘鎵绀
