Is this function a probability density function?

Question

$$ f(x,y) = \begin{cases} \dfrac{6}{10^5} x^2 y & \text{$0<x<10$, $0<y<10$} \\[1ex] 0 & \text{otherwise} \end{cases} $$

1. is this a PDF function since double integral with limits sums to 1?
2. If 1 is True, can we say $x$ and $y$ (within support) are then random variables?

I would appreciate you reply.

P.s. I almost 40, I don't go to school anymore and I am not doing school homework. It's based on my interest.:))

Answer 1

This is indeed a pdf, since, as you already stated, the integral over the domain $\mathbb{R}^2$ is 1 and it is non-negative.

Calling $x$ and $y$ random variables is not correct. The answer is a bit more subtle, namely, you might define two random variables $X$ and $Y$ that have joint density $f.$ A random variable is a function from the sample space to $\mathbb{R}.$ $x$ and $y$ here are just variables, not random variables. They are the arguments of the density function $f.$