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
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asked 2021-05-08
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.

Expert Answers (1)

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
\(f_{Y}(y)=\int_{-\infty}^{\infty}f(x,y)dy\)
\(=\int_{0}^{1}\frac{1}{3}(2x+3y)dy\)
\(=\frac{1}{3}\int_{0}^{1}(2x+3y)dy\)
\(=\frac{1}{3}[2xy+\frac{3y^{2}}{2}]_{0}^{1}\)
\(=\frac{1}{3}[(2x+\frac{3}{2})-0]\)
\(=\frac{4x+3}{6}\)
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