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# 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 X.

Random variables
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asked 2021-06-05
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 X.

## Answers (1)

2021-06-06
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 x
$$f_{X}(x)=\int_{-\infty}^{\infty}f(x,y)dx$$
$$=\int_{0}^{1}\frac{1}{3}(2x+3y)dx$$
$$=\frac{1}{3}\int_{0}^{1}(2x+3y)dx$$
$$=\frac{1}{3}[x^{2}+3xy]_{0}^{1}$$
$$=\frac{1}{3}[(1+3y)-0]$$
$$=\frac{3y+1}{3}$$

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