I'm reading the book All of statistics of Larry Wasserman, on the exercise 3.32 is about independent random variables, all seems fine for me, but I can't understand why in the second integral, goes from $0$ to $1-x$, where do we get $1-x$?
2026-04-03 09:57:48.1775210268
why the integral of a joint function goes from $0$ to $1-x$
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Note that $$ x+y\leq 1\implies y\leq 1-x $$ Combined with the fact that $0\leq x\leq 1$ and $0\leq y\leq 1$, we can write the domain of integration as $(x,y)$ pairs such that $$ 0\leq y\leq 1-x;\quad \text{and} \quad0\leq x\leq 1. $$ Here is a picture of the domain of integration shaded in green.