Find distribution function and density of random variable

Question

Point is selected uniformly from the shape $1 \leq |x| + |y| \leq 3, y > 0$.

Find distribution function and density of random variable $X = x$.

I find that $x \in [-3; 3]$ and $y \in (0,3]$, how can i find distribution function?

Answer 1

HINT

In this uniformly drawn point, let $X$ denote the first coordinate and $Y$ denote the second one.

• Plot the region on which $X,Y$ are defined.
• Find the joint distribution function $f_{XY}$ of $X,Y$
• You can compute the marginal distribution of $X$ as $$ f_X(x) = \int_\mathbb{R} f_{XY}(x,y)\ dy. $$