Question

设二维随机变量(x,y)的概率密度为f(x,y)=e的-x次方？

急急急！！！

Answer 1

如图所示

Answer 2

(1)
Z=X+Y
F(z)=P(Z<z)=P(X+Y<z)=∫(0,z/2)∫(x,z-x) f(x,y) dydx
=-2e^(-z/2)+1+e^(-z)
fz(z)=F'(z)=e^(-z/2)-e^(-z)
(2)
fX|Y(x|y)=f(x,y)/fY(y)
Y的边缘密度是fY(y)=∫(0,y) e^(-y)dx=ye^(-y)
所以fX|Y(x|y)=1/y
(3)
P{X>3|Y<5)=P(X>3 Y<5)/P(Y<5)
P(X>3 Y<5)=∫(3,5)∫(x,5) e^(-y)dydx=e^(-3)-3e^(-5)
P(Y<5)=∫(0,5) ye^(-y)dy=1-6e^(-5)
所以P{X>3|Y<5)=(e^2-3)/(e^5-6)
(4)
(2)已求出fX|Y(x|y)=1/y
所以fX|5(x|5)=1/5
P{X>3︱Y=5}=∫(3,5) 1/5dx=2/5

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