# Probability and Covariance given joint pdf

Let $X$ and $Y$ be random variables with joint PDF $f(x,y) = 2$ for $0 ≤ x ≤ y ≤ 1$, and $0$ elsewhere.

a) Find $Pr(X +Y ≤ 1)$

b) Find covariance of $X$ and $Y$

I'm not really sure how to find that probability...the bounds are confusing me a bit. Similar with the covariance. Are the bounds just the $1\times1$ square?

The joint support, $\{(x,y):0\leq x\leq y\leq 1\}$, is a right-triangle.   $\triangle(0,0)(0,1)(1,1)$ to be precise.
As they are jointly uniform distributed, the probability may be found directly by geometry -- just sketch a graph of where $y\leq 1-x$ intercepts the triangle (a smaller triangle) and compare.   You might also use this to determine bounds for an integration, for practice.
For the other part, the covariance will be determined by integrating the relevant functions over the given support. $$\displaystyle\int_0^1\!\!\int_0^y \ldots~\mathrm d x\mathrm d y$$
• The probability is measured over: $0\leq x\leq 1$ , $0\leq y\leq 1-x$, and $x\leq y\leq 1$ . The intersection of the event description and the support. – Graham Kemp Jun 27 '18 at 14:47
• No, the correlation is positive because $\mathsf EX$ and $\mathsf EY$ are smaller than that. (Reality check: do you expect $Y$ to lie on the edge of the support?) – Graham Kemp Jun 27 '18 at 14:50
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