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,

\(f(x)=\int_{0}^{1}f(x,y)dy\)

\(=\int_{0}^{1}8xydy\)

\(=8x[\frac{y^{2}}{2}]_{0}^{1}\)

\(=4x\)