Probability of normal distribution larger than half normal distribution

Question

I am pretty new to normal distribution and stuck on this question:

X ~ N(0, 1)

Y ~ N(0, 1)

X independent of Y

What is the probability of P(X>|2Y|)?

I know linear combination of normal distribution is normal distribution but what about linear combination of normal distribution and absolute value of normal distribution?

Answer 1

You can view $(X,Y)$ as a random point on the plane $\mathbb{R}^2$. The joint distribution is the standard bivariate normal distribution and is radially symmetric about the origin.

Plot the region consisting of pairs $(x,y)$ satisfying $x > |2y|$; you will find this is an infinite cone (an infinite slice of pizza or pie, if you will) with vertex at the origin. By the symmetry mentioned above, the probability of the random vector $(X,Y)$ being in this slice is equal to the relative angle of the slice (relative to a full turn, i.e. $2\pi$ radians or $360$ degrees).