What is the expected value of a standard normal random variable given value is positive?

Question

Am not sure if I'm wording this correctly.
But say we take huge sample of standard normal random variables. Then we separate out positive values. What would be average of the positive values ? What would it approach as sample size got larger and larger.

Answer 1

The density is $$ f(x) = \begin{cases} \frac2{\sqrt{2\pi}} e^{-x^2/2} & \text{if }x>0, \\[6pt] 0 & \text{if }x<0. \end{cases} $$ So the expected value is $$ \int_0^\infty xf(x)\,dx = \frac2{\sqrt{2\pi}} \int_0^\infty e^{-x^2/2} \big(x\,dx\big) = \frac2{\sqrt{2\pi}} \int_0^\infty e^{-u}\,du = \frac2{\sqrt{2\pi}} \cdot 1 = \sqrt{\frac{2}{\pi}}. $$

Answer 2

$$ \begin{align} P(x=x_0 \mid x>0) &= \frac{P(x=x_0)}{P(x>0)} \\ &= \frac{\frac{1}{\sqrt{2 \pi}} e^{-x_0^2/2}}{\int_0^\infty \frac{1}{\sqrt{2 \pi}} e^{-z^2/2} dz} \\ &= \sqrt{\frac{2}{\pi}} e^{-x_0^2/2} \end{align} $$

So

$$ E(x \mid x>0) = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-z^2/2} z\, dz = \sqrt{\frac{2}{\pi}} $$