\sqrt{3}X)$ if $(X,Y)$ are i.i.d. standard normal

Asked Apr 11 '17 at 21:07

Active Apr 11 '17 at 21:21

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Let $X$ and $Y$ be independent standard normal random variables. Find:
i) $P(X > 0, Y > 0)$
ii) $P(X > 0, Y > X)$
iii) $P(X > 0, Y > \sqrt{3}X)$

I am new to this, but I think I have the geometry down. This should involve angles, correct? I interpret this as the relevant volume under the joint density surface as a quadrant-shaped slice of a bell-shaped object. Can someone walk me through the solutions to these?

edited Apr 11 '17 at 21:21

Did

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asked Apr 11 '17 at 21:07

Jayant.M

3

Hint (a huge one): $(X,Y)=(R\cos\Theta,R\sin\Theta)$ where $R>0$ almost surely, $(R,\Theta)$ is independent, and $\Theta$ is uniform on $[0,2\pi)$. Apply this. – Did Apr 11 '17 at 21:22
do you have a hint for how to apply that to part (i) for example? – Jayant.M Apr 12 '17 at 01:30
i) is trivial since ${X>0}$ and ${Y>0}$ are independent, each with probability $\dfrac{1}{2}$. – ChargeShivers Apr 12 '17 at 01:41
"do you have a hint for how to apply that" Sure: translate each of the three events involved in terms of $\Theta$. – Did Apr 12 '17 at 06:53
https://math.stackexchange.com/q/255368/321264 – StubbornAtom Oct 12 '21 at 14:29

Find $P(X > 0, Y > 0)$, $P(X > 0, Y > X)$ and $P(X > 0, Y > \sqrt{3}X)$ if $(X,Y)$ are i.i.d. standard normal

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