Question

# Two random variables X and Y with joint density function given by: f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases} Find the marginal density of Y.

Random variables
Two random variables X and Y with joint density function given by:
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$
Find the marginal density of Y.

2021-05-09
The joint density function of random variables X and Y is :
$$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & otherwise\end{cases}$$
We have to find :
marginal density of y
$$f_{Y}(y)=\int_{-\infty}^{\infty}f(x,y)dy$$
$$=\int_{0}^{1}\frac{1}{3}(2x+3y)dy$$
$$=\frac{1}{3}\int_{0}^{1}(2x+3y)dy$$
$$=\frac{1}{3}[2xy+\frac{3y^{2}}{2}]_{0}^{1}$$
$$=\frac{1}{3}[(2x+\frac{3}{2})-0]$$
$$=\frac{4x+3}{6}$$