Question

The joint PDF of the random variables f(x,y)=8xy,0<x<1, 0<y<x

f(x,y)=8xy,0<x<1, 0<y<x. Find the marginal probability density function of Y.

Answer 1

At first, we check that $\iint_{{\mathbb{R}}^2}f(x,y)dxdy=1$. $\iint_{{\mathbb{R}}^2}f(x,y)dxdy=\int_0^1\int_{0}^x8xydydx=\int_0^14{x^3}dx=x^4|_0^1=1$. The marginal probability density of $Y$ is: $f_Y(y)=\int_{\mathbb{R}}f(x,y)dx=\int_y^18xydx=4yx^2|_y^1=4(y-y^3)$.

Answer: the marginal probability density function of $Y$ is: $f_Y(y)=4(y-y^3).$