$\mathbb{P}(|X|<1,|Y|<2)$ When $X,Y$ Are I.I.D. Standard Normal

Question

Calculate $\mathbb{P}(|X|<1,|Y|<2)$ when $X,Y$ are i.i.d. standard normal r.v.s.

I think the answer is simply $$(\Phi(1)-\Phi(-1))(\Phi(2)-\Phi(-2)).$$

Is this correct? Thanks.

Answer 1

yes, since your variables are i.i.d $$\mathbb{P}(|X|<1,|Y|<2) = \mathbb{P}(|X|<1) \mathbb{P}(|Y|<2)$$ and you've correctly identified $\mathbb{P}(|X|<1) = \mathbb{P}(X<1) - \mathbb{P}(X<-1)$. similarly with $Y$