1 Let X1 and X2 have joint density fx1x2 8x1x210 leq x1 Sol

1. Let X1 and X2 have joint density f(x1,x2) = 8x1x21[0 leq x1

Solution

a) f( 1,1) = 8

f(x1 = 1) = 4

f(x2=1) =4

As f(1,1) not equal to f(1) f(2) x1 and x2 are not independent.

x1 = y1y2 and x2 =y2

J = y2 y1

0 1 = y2

Hence pdf of (y1,y2) = 8y1y23, 0<=y1<=y2<=1 ( as total prob =1)

c) f(y1) = 2 y1,0<=y1<=1

d) f(y1*y2) = f(x1)

Thus it is purely independent.