Question

Let (𝑋, 𝑌, 𝑍) be a random vector having the joint probability density function
$f(x,y, z) =\begin{cases} \frac{1}{2\,xy}, & \quad \text{if }0<z<y<x<1.\\\ \frac{1}{2x^2}, & \quad if\,0<z<x<y<2x<2,\\\0, & \quad\,\,Otherwise.\end{cases}$
Then, which one of the following statements is FALSE?

• A. $𝑃(𝑍 < 𝑌 < 𝑋) = \frac{1}{2}$
• B. 𝑃(𝑋 < 𝑌 < 𝑍) = 0
• C. 𝐸(min{𝑋, 𝑌}) =$\frac{1}{4}$
• D. 𝑉𝑎𝑟 (𝑌 | 𝑋 = $\frac{1}{2}$ ) = $\frac{1}{12}$

Answer 1

Answer: (C)

𝐸(min{𝑋, 𝑌}) =$\frac{1}{4}$

Answer 2

The correct option is (C): 𝐸(min{𝑋, 𝑌}) =$\frac{1}{4}$