Check if $X$ and $Y$ are statistically independent from a pdf

Question

Consider the function $f(x,y)=2-x-y$ on the rectangular area $0\leq x\leq 1$ ; $0\leq y\leq1$ and $0$ otherwise.

1. Show that $f(x,y)$ is a probability density function.
2. Check whether $X$ and $Y$ are statistically independent.

I've had trouble showing/proving that a single variable pdf is a actually a pdf, but my real question is how to check whether $X$ and $Y$ are statistically independent.

Answer 1

For part 1, check that $$ \int_0^1 \int_0^1 (2-x-y) \, \mathrm{d}y \, \mathrm{d}x = \int_0^1 \left(2-x-\frac{1}{2} \right) \, \mathrm{d}x = 2- \frac{1}{2} -\frac{1}{2} = 1. $$

For part 2, check whether $f(x,y) = g(x) h(y)$. Since your $f(x,y)$ does not appear to be separable...