Question

Let X and Y have the continuous joint distribution with p.d.f.

f(x,y)\left\{\begin{matrix} \frac{1}{\pi a^2,} if x^2 + y^2\leq a^2\\ 0, otherwise \end{matrix}\right.

where a > 0.Compute the marginal densities of X and Y. Check whether X and Y are independent.

14 Feb 2018
Answer :
Word Count : 522
To solve this problem, we are given the joint probability density function (p.d.f.): \[ f(x, y) = \left\{ \begin{matrix} \frac{1}{\pi a^2}, & \text{if } x^2 + y^2 \leq a^2 \\ 0, & \text{otherwise} \end{matrix} \right. \] ### Step 1: Marginal density of X The marginal density of \(X\), denoted as \(f_X(x)\), is obtained by integrating the joint p.d.f. over all possible values of \(y\). That is: \[ f_X(x) = \int_{-\infty}^{\infty} f(x, y) \, dy \] Since \(f(x, y)\) is nonzero only when \(x^2 + y^2 \leq a^2\), we need to find the bounds on \(y\) for a given \(x\). The constraint \(x^2 + y^2 \leq a^2\) gives: \[ y^2 \leq a^2 - x^2 \quad \Rightarrow \quad -\sqrt{a^2 - x^2} \leq y \leq \sqrt{a^2 - x^2} \] Thus, the marginal density of \(X\) is: \[ f_X(x) = \int_{-\sqrt{a^2 - x^2}}^{\sqrt{a^2 - x^2}} \frac{1}{\pi a^2} \, dy \] The integral can be computed: \[ f_X(x) = \frac{1}{\pi a^2} \left( \sqrt{a^2 - x^2} - (-\sqrt{a^2 - x^2}) \right) \] \[ f_X(x) = \frac{1}{\pi a^2} \times 2\sqrt{a^2 - x^2} \] \[ f_X(x) = \frac{2\sqrt{a^2 - x^2}}{\pi a^2}, \quad \text{for } -a \leq x \leq a \] For \(x\) outside this range (\(x < -a\) or \(x > a\)), \(f_X(x) = 0\). ### Step
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