Question

Let \((X,Y)\) be a random vector with the joint probability density function

• A. The marginal distribution of \(X\) is \(Exp(1)\)
• B. \(E(Y|X=4)=0.25\)
• C. \(E(XY)=1.5\)
• D. \(E(Y)\) is not finite

Hint:

For joint densities, first find the marginal density, then use \(f_{Y|X}(y|x)=\frac{f(x,y)}{f_X(x)}\) to identify the conditional distribution.

Answer 1

The Correct Option is A, B, D

Step 1: Marginal of $X$ (A).
$f_X(x)=\int_0^\infty xe^{-x(y+1)}dy=xe^{-x}\cdot\frac1x=e^{-x}$, so $X\sim Exp(1)$. (A) holds.

Step 2: Conditional of $Y$ (B).
$f_{Y|X}(y|x)=\frac{xe^{-x(y+1)}}{e^{-x}}=xe^{-xy}$, so $Y|X=x\sim Exp(x)$ with mean $1/x$. At $x=4$ this is $0.25$, so (B) holds.

Step 3: $E(XY)$ (C).
$E(XY)=E\{X\cdot E(Y|X)\}=E(X\cdot\frac1X)=1\ne1.5$. (C) fails.

Step 4: $E(Y)$ (D).
$E(Y)=E(1/X)=\int_0^\infty\frac1x e^{-x}dx$, which diverges near $0$. So $E(Y)$ is not finite, and (D) holds.

Step 5: Collect.
\[ \boxed{(A),(B),(D)} \]