第1个回答 2012-05-03
(a) Sketch the graph of the support of X and Y .
omitted
(b) Find f1(x), the marginal pdf of X.
f1(x)=∫【0，x】8xydy=4x^3,0≤x≤1
(c) Find f2(y), the marginal pdf of Y
f2(y)=∫【y，1】8xydx=4y-4y^3,0≤y≤1
"(d) Find the mean of X, mean of Y, variance of X, variance of Y, Cov(X,Y), ρ
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EX=∫【0，1】x*f1(x)dx=4/5
EX^2=∫【0，1】x^2*f1(x)dx=2/3
DX=EX^2-(EX)^2=2/75
EY=∫【0，1】y*f2(y)dy=8/15
EY^2=∫【0，1】y^2*f1(y)dy=1/3
DY=EY^2-(EY)^2=11/225
EXY=∫∫xyf(x,y)dxdy=4/9
Cov(X,Y)=EXY-EX*EY=4/225
ρ=Cov(X,Y)/√(DX*DY)=2√110/55
(e) Find g(x|y = 1/2), the conditional pdf of X given Y = 1/2.
g(x|y )=f(x,y)/f2(y)=2x/(1-y^2)
g(x|y=1/2 )=8x/3
(f) Find P(X<3/4 | Y=1/2).
P(X<3/4 | Y=1/2)=∫【0，3/4】g(x|y=1/2 )dx=3/4
(f) Find P(X<3/4 | Y<1/2).
P(X<3/4 | Y<1/2)=P(X<3/4, Y<1/2)/P(Y<1/2)
P(X<3/4, Y<1/2)=∫【0，1/2】dy∫【y，3/4】f(x,y)dx=7/32
P(Y<1/2)=∫【0，1/2】f2(y)dy=7/16
P(X<3/4 | Y<1/2)=P(X<3/4, Y<1/2)/P(Y<1/2)=1/2
解毕