Question

Let X

1
​
and X
2
​
have the joint pdf f
X
1
​
,X
2
​

​
(x
1
​
,x
2
​
)=2e
−x
1
​
−x
2
​

I
(0,x
2
​
)
​
(x
1
​
)I
(0,[infinity])
​
(x
2
​
). (a) Find the marginal pdfs for X
1
​
and X
2
​
. (b) Are X
1
​
and X
2
​
independent? (c) Suppose that Y
1
​
=2X
1
​
and Y
2
​
=X
2
​
−X
1
​
. Show that Y
1
​
and Y
2
​
are independent.

Answer 1

Answer:(a) To find the marginal pdf of X1, we integrate f(X1,X2) over all possible values of X2:

fX1(x1) = ∫[0,∞] 2e^(-x1-x2) dx2

= -2e^(-x1-x2)|[0,∞]

= 2e^(-x1)

Similarly, to find the marginal pdf of X2, we integrate f(X1,X2) over all possible values of X1:

(b) To check whether or not X1 and X2 are independent, we need to compare their joint distribution with the product of their marginal distributions. That is,