Is the variance of these two variables the same?

Question

Let $X$ and $Y$ be random variables having joint density function $$ f(x,y) = \begin{cases} x + y & \text{for } 0 \leq x \leq 1, 0 \leq y \leq 1 \\ 0 & \text{other }x, 0 \leq y \leq 1 \end{cases} $$

In this case, I claim that the $Var(x) = Var(y)$ because the density function is symmetric. Is my claim correct?

Note: I am not looking for somebody to tell me how to find $Var(x)$. I do realize that: $$ Var(x) = \mathbb{E}\left[x^2\right] - (\mathbb{E}[x])^2 $$

Answer 1

1. You likely mean a more general second condition on the pdf and
2. if so, the problem is symmetric in $x,y$ so indeed the variances (and any other characteristics) would be identical.