# Average of a tail of a normal distribution

I have a sample that is pretty normally distributed with a mean greater 0. Is there any way to get a closed form estimate for the mean of all values that are less than 0? In other words, given a mean, variance and a variable x, can I estimate the average of all values that are smaller than x?

• Not in the usual sense of closed form; if you admit the use of the cdf of a standard normal, then, the answer is yes. – cardinal Apr 20 '12 at 1:14
• – Cyan Apr 20 '12 at 1:17

Cyan offered a link that answers the question but since the question remains without an actual answer, I'll put one in.

While not strictly "closed form" by the usual definitions, I expect people doing statistical work will mostly want to admit the normal cdf (or the error function, which would serve the same purpose) into what they call "closed form".

Given $X\sim N(\mu,\sigma^2)$,

$$E(X\mid X<b)=\mu -\sigma \frac {\phi (\frac{b-\mu}{\sigma})}{\Phi (\frac{b-\mu}{\sigma})}$$

where $\phi$ is the standard normal density and $\Phi$ is the standard normal cdf.

For the present problem, $b=0$, giving us:

$$E(X\mid X<0)=\mu -\sigma \frac {\phi (\frac{-\mu}{\sigma})}{\Phi (\frac{-\mu}{\sigma})}$$

(An expression for the variance can be found at the above link.)