Word Count : 519

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We are given the joint density function of random vector $$ \underline{X} = \begin{pmatrix} X_1 \\ X_2 \end{pmatrix}, \quad \text{with} \quad f(x_1, x_2) = \begin{cases} 4x_1x_2, & 0 < x_1 < 1,\ 0 < x_2 < 1 \\ 0, & \text{otherwise} \end{cases} $$ --- ### Step 1: Marginal Distributions Marginal density of $X_1$: $$ f_{X_1}(x_1) = \int_0^1 f(x_1, x_2)\, dx_2 = \int_0^1 4x_1x_2\, dx_2 $$ $$ = 4x_1 \int_0^1 x_2\, dx_2 = 4x_1 \left[ \frac{x_2^2}{2} \right]_0^1 = 4x_1 \cdot \frac{1}{2} = 2x_1 $$ $$ \Rightarrow f_{X_1}(x_1) = \begin{cases} 2x_1, & 0 < x_1 < 1 \\ 0, & \text{otherwise} \end{cases} $$ Marginal density _______ _____ ___ ________ ________ _____.
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