Random variable X and Y have the joint PDFf_{XY}(x,y)=\begin{cases}c,\ x

Question

Random variable X and Y have the joint PDF
$f_{X Y} (x, y) = {\begin{cases} c, x \geq 0, y \geq 0, (x^{2} + y) \leq 1, \\ 0, o t h e r w i s e \end{cases}$
Find the marginal PDF $f_{Y} (y)$ .

Answer 1

The marginal PDF $f_{Y} (y)$ is obtained as given below:
$f_{X, Y} (x, y) = {\begin{matrix} \frac{3}{2}; x \geq 0, y \geq 0, (x^{2} + y) \leq 1 \\ 0; o t h e r w i s e \end{matrix}}$
$f (y) = \int_{x} f (x, y) d x$
$= \int_{0}^{\sqrt{1 - y}} \frac{3}{2} d x$
$= {[\frac{3}{2} x]}_{0}^{\sqrt{1 - y}}$
$= \frac{3 \sqrt{1 - y}}{2}$
$f (y) = \frac{3 \sqrt{1 - y}}{2}; 0 \leq y \leq 1$

Random variable X and Y have the joint PDFf_{XY}(x,y)=\begin{cases}c,\ x

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