Question

Let the joint probability density function of the random variables 𝑋 and 𝑌 be
$f(x,y) = \begin{cases} 1, & \quad 0<x<1,\,\,\,\,x<y<x+1\\\ 0,& \quad \ \text{ Otherwise} \end{cases}$
Let the marginal density of 𝑋 and 𝑌 be 𝑓𝑋(𝑥) and 𝑓𝑌 (𝑦), respectively. Which of the following is/are CORRECT?

• A. $f_x(x) = \begin{cases} 2x,\,\,\,0<x<1 & \quad \\\ 0, \,\,\,\,\text{Otherwise}\end{cases}$ And $f_y(y) = \begin{cases} 2-y,\,\,\,0<y<2 & \quad \\\ 0, \,\,\,\,\text{Otherwise}\end{cases}$
• B. $f_x(x) = \begin{cases} 1,\,\,\,0<x<1 & \quad \\\ 0, \,\,\,\,\text{Otherwise}\end{cases}$ And $f_y(y) = \begin{cases} y,\,\,\,\,\,\,\,\,\,\,\,\,0<y<1 & \quad \\\ 2-y, \,\,1≤y<2\\\0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Otherwise}\end{cases}$
• C. $E(X)=\frac{1}{2}$, var(X)=$\frac{1}{12}$
• D. $E(Y)=1$, var(Y)= $\frac{1}{6}$

Answer 1

Answer: (B), (C), (D)

$f_x(x) = \begin{cases} 1,\,\,\,0<x<1 & \quad \\\ 0, \,\,\,\,\text{Otherwise}\end{cases}$ And $f_y(y) = \begin{cases} y,\,\,\,\,\,\,\,\,\,\,\,\,0<y<1 & \quad \\\ 2-y, \,\,1≤y<2\\\0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Otherwise}\end{cases}$

Answer 2

The correct options is (B): $f_x(x) = \begin{cases} 1,\,\,\,0<x<1 & \quad \\\ 0, \,\,\,\,\text{Otherwise}\end{cases}$ And $f_y(y) = \begin{cases} y,\,\,\,\,\,\,\,\,\,\,\,\,0<y<1 & \quad \\\ 2-y, \,\,1≤y<2\\\0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Otherwise}\end{cases}$, (C): $E(X)=\frac{1}{2}$, var(X)=$\frac{1}{12}$ and (D): $E(Y)=1$, var(Y)= $\frac{1}{6}$