Question

The joint density of the random variables X and Y is given by f(x,y)=\begin{cases}8xy & 0\leq x\leq 1, 0\leq y \leq x\\0 & otherwise\end{cases} Find the marginal density of X

Random variables
ANSWERED
asked 2021-06-02
The joint density of the random variables X and Y is given by
\(f(x,y)=\begin{cases}8xy & 0\leq x\leq 1, 0\leq y \leq x\\0 & otherwise\end{cases}\)
Find the marginal density of X

Expert Answers (1)

2021-06-03

Step 1
We have been given the joint density function of random variables X and Y as,
\(f(x,y)=\begin{cases}8xy & 0\leq x\leq 1, 0\leq y \leq x\\0 & otherwise\end{cases}\)
\(=\begin{cases}8xy & 0\leq x\leq 1, 0\leq y \leq x\\0 & otherwise\end{cases}\)
The marginal density function of X is computed as,
\(f(x)=\int_{0}^{1}f(x,y)dy\)
\(=\int_{0}^{1}8xydy\)
\(=8x[\frac{y^{2}}{2}]_{0}^{1}\)
\(=4x\)

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