IGNOU MST 3 2018 - The joint density function of random variables X and Y is given by &nb

Question

The joint density function of random variables X and Y is given by

$f(x,y) = 4xye - ^{(x^2 + y^2)},x \geq 0,\geq 0$

Are X and Y independent?

16 Feb 2018

Answer :

Word Count : 339

We are asked to determine whether $X$ and $Y$ are independent given the joint density: $$ f(x,y) = 4xy e^{-(x^2 + y^2)}, \quad x \ge 0, \ y \ge 0. $$ Let's solve this step by step manually. --- ### Step 1: Recall the criterion for independence Two continuous random variables $X$ and $Y$ are independent if and only if: $$ f(x,y) = f_X(x) \cdot f_Y(y), $$ where $f_X(x)$ and $f_Y(y)$ are the marginal densities of $X$ and $Y$. --- ### Step 2: Find the marginal density of $X$ $$ f_X(x) = \int_0^\infty f(x,y) \, dy = \int_0^\infty 4xy e^{-(x^2 + y^2)} \, dy $$ Factor out $x$ and $e^{-x^2}$ since they don’t depend on $y$: $$ f_X(x) =

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