Albarellak

2021-10-21

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

SchulzD

The joint density function of random variables X and Y is :
$f\left(x,y\right)=\left\{\begin{array}{ll}\frac{1}{3}\left(2x+4y\right)& 0\le x,\le 1\\ 0& otherwise\end{array}$
We have to find :
marginal density of y
${f}_{Y}\left(y\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f\left(x,y\right)dy$
$={\int }_{0}^{1}\frac{1}{3}\left(2x+3y\right)dy$
$=\frac{1}{3}{\int }_{0}^{1}\left(2x+3y\right)dy$
$=\frac{1}{3}{\left[2xy+\frac{3{y}^{2}}{2}\right]}_{0}^{1}$
$=\frac{1}{3}\left[\left(2x+\frac{3}{2}\right)-0\right]$
$=\frac{4x+3}{6}$

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