Let X be a Gaussian random variable with mean 0 and variance sigma^2. Let Y = max(X, 0) where max(a, b) is the maximum o...

Question

Let X be a Gaussian random variable with mean 0 and variance sigma^2. Let Y = max(X, 0) where max(a, b) is the maximum of a and b. The median of Y is ____________.

Accepted Answer

The median of Y = max(X, 0) is 0. Y = max(X, 0) is the positive part of a zero-mean Gaussian. When X < 0, Y = 0; when X >= 0, Y = X. Since X is symmetric around 0, exactly half the time X < 0 (Y = 0) and half the time X >= 0 (Y = X >= 0). The median is the value m such that P(Y <= m) = 0.5. Because P(Y = 0) = P(X <= 0) = 0.5, the median is 0.