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 𝑓_𝑋(𝑥) and 𝑓_𝑌 (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

The Correct Option is B, C, D

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)= \(\frac{1}{6}\)