# Expected value of normal distribution given that distribution is positive

Given $X \sim N(0, \sigma^2)$ (that is, $X:\mathbb{R} \to \mathbb{R}$ is a normal random variable with mean $0$ and variance $\sigma^2$), I'm trying to calculate the expected value of $X$ given that $X>0$. I thought that integrating $$\int_{0}^{\infty} x\cdot \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{x^2}{2\sigma^2}}dx$$ would do it, but the value, $\frac{\sigma}{\sqrt{2\pi}}$, seems to be off by a factor of 2 based on some other information I have; I think the answer should be $\sqrt{\frac{2}{\pi}}\sigma$.

Question: How should the expected value of $X$, given that $X>0$, be computed?

• You computed $E(X\mathbf 1_A)$, where $A=[X\gt0]$, instead of $E(X\mid A)$. – Did Dec 3 '11 at 22:13
• Wolfram gives the same answer as what you get. Why do you think your answer is off by factor of 2? – tards Dec 3 '11 at 22:14
• @tards I know the computation of the integral is correct, I'm just don't think that the integral represents what I actually want, as Michael Lugo's answer confirms. – Quinn Culver Dec 3 '11 at 22:23
• "Exact duplicate"? I think I posted the answer to this same question about two months or so ago. – Michael Hardy Dec 3 '11 at 22:38
• @MichaelHardy I don't see how the post you liked is a duplicate. Please explain. – Quinn Culver Dec 4 '11 at 14:54

## 1 Answer

Let $f(x)$ be the density of $X$; let $F(x)$ be its CDF.

Then the density of $X$, conditional on it being positive, is $f(x)/P(X \ge 0)$ if $x \ge 0$, and $0$ otherwise.

Of course $P(X \ge 0) = 1/2$ by symmetry, so the density of $X$ conditional on $X \ge 0$ is $2f(x)$ (on $x \ge 0$).

So you need to do the integral $$\int_0^\infty 2xf(x) \: dx$$ which is twice the integral you've written.

• Also note, $E[X|H] = \frac{E[1_HX]}{P(H)}$. Hence, $E[X|X>0] = \frac{E[1_HX]}{P(X>0)}$ which will give you the desired integral to evaluate with the factor 2 in front. – Gabor Bakos Mar 23 at 16:06