Question

The joint density of random variable X and Y is $f_{XY}(x,y)=\begin{cases}2x & \text{for } 0<x<1, x<y<x+1 \\ 0 & \text{otherwise}\end{cases}$ then marginal of Y is

• A. $f_{Y}(y)=\begin{cases}y^2 & 0<y<1
  y(2-y) & 1<y<2
  0 & \text{otherwise}\end{cases}$
• B. $f_{Y}(y)=\begin{cases}1 & 0<y<1
  0 & \text{otherwise}\end{cases}$
• C. $f_{Y}(y)=\begin{cases}2y & 0<y<1
  0 & \text{otherwise}\end{cases}$
• D. $f_{Y}(y)=\begin{cases}3y^2 & 0<y<1
  0 & \text{otherwise}\end{cases}$

Hint:

When the bounds of one variable depend on another, always sketch the region or use the $\max/\min$ logic to split your integral into cases. Marginal densities in such cases are often piecewise functions.

Answer 1

The Correct Option is A