Re: mean and standard deviation of left-truncated normal distribution

From: Ray Koopman (koopman_at_sfu.ca)
Date: 03/22/05 

Date: 21 Mar 2005 16:27:12 -0800

clemenr@wmin.ac.uk wrote:
> Hi. I would like to know the mean and standard deviation of a trunc-
> ated normal distribution, calculated from the mean and sd of the
> underlying normal distribution, and the (left) truncation point k.
>
> Clearly I can estimate the mean and sd given a simulation, which
> in reality will be amply good enough for my purposes, but I feel
> that it should be easy to calculate the true mean and sd.
>
> I did find a paper with formulae for the mean and sd of a
> standardised truncated normal distribution (underlying mean 0,
> underlying sd 1, truncation point k).
>
> However, I'm not 100% sure how to convert these into mean and
> standard deviation for an arbitrary left-truncated normal
> distribution. I would expect this to be a simple proces,
> but I'm not quite confidence as to how it should be done.

If a standard normal distribution is left-truncated at k then the
mean becomes m = f/Q, where f is the density at k and Q is the
area to the right of k, and the variance becomes s^2 = 1 - m(m-k).

For an aribitrary normal distribution, with mean M, variance S^2,
and truncation point K, get m and s as above using k = (K-M)/S.
To get back to the original units, use m -> m*S + M and s -> s*S.

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