Question: Let a joint density function $f_{X,Y}: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by:
\begin{equation} f_{X,Y}(x,y) = \begin{cases} cx^2y \quad &\text{if }0<x<1 \text{ and max}\{b,x\} <y <2\\ 0 &\text{else} \end{cases} \end{equation} where $b\in[0,1]$.
We take for granted that for every $b$, there exists some constant $c \in \mathbb{R}$ such that $f_{X,Y}$ is a valid joint density function.
Are random variables X and Y independent?
My solution: So the solution was independent for $b = 1$, and dependent for $b < 1$. I only managed to solve it for $b = 1$.
I solved it by finding the value $c$ which makes the above a valid density (i.e. integral over $\mathbb{R}^2 = 1$), computing the marginals $f_X$ and $f_Y$, and showing that $f_{X,Y} = f_X f_Y$ which is true iff X and Y are independent.
This was simple to compute because the bounds for integration were clear. i.e. $0<x<1, \quad 1<y<2$.
However, I'm not sure how to solve it for $b < 1$. I attempted to use the same approach of computing the marginals, but I couldn't figure out a way to make the bounds of integration work.
i.e. I wasn't sure how to compute:
$f_X = \int^{\infty}_{-\infty} f_{X,Y} dy, \quad f_Y = \int^{\infty}_{-\infty} f_{X,Y} dx, \quad$ given the bounds $0<x<1 \text{ and max}\{b,x\} <y <2$.
Is there a clean way to compute the above integrals? My multivariable calculus is a little rusty. Thank you!
Given that there is a joint density, the variables are independent iff this joint density can be written as a product: $f_{XY}=h(x)g(y)$ for a.e. $x,y \quad$(*)
Indeed, given (*) you can easily deduce that $h=f_X$ and $g=f_Y$.
This settles the case $b=1$.
For $b<1$, check that both events $\{X>(b+1)/2\}$ and $\{Y<(b+1)/2\}$ have positive probability, yet their intersection has probability $0$.