Mean of a standard normal given it is positive

Question

For $X\sim N(0,1)$ , compute $\mathbb{E}[X\mid X>0]$ .

Short answer

$\sqrt{2/\pi}\approx0.798$ . $\mathbb{E}[X\mathbf 1_{X>0}]=\int_0^\infty x\,\tfrac{1}{\sqrt{2\pi}}e^{-x^2/2}dx=\tfrac{1}{\sqrt{2\pi}}$ , and dividing by $\mathbb{P}(X>0)=\tfrac12$ gives $\sqrt{2/\pi}$ .

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 3.2 Integration, 'Expected value using integration', p.39; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

The Normal Distribution
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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