variance of doubly truncated binomial

Question

I am in need of variance of doubly truncated binomial distribution (equation number 3.69 on page number 137 of third edition of Johnson, Kemp and Kotz Discrete Probability Distributions).

Thanks. Anwer

Answer 1

Let $X \sim \text{Binomial}(n,p)$ with pmf $f(x)$:

Let $a$ and $b$ denote the lower and upper points of truncation, such that $a \leq X \leq b$.

Then, $P(a \leq X \leq b)$ is:

where I am using the Prob function from the mathStatica package for Mathematica to do the nitty-gritties.

Then, the pmf of the doubly truncated random variable $X$ s.t. $a \leq X \leq b$, with pmf denoted $g(x)$, is:

Finally, we seek $\text{Var}_g(X)$:

All done. The solution output can be viewed LARGER here:

http://www.tri.org.au/se/varianceofdoublytruncatedBinomial.png

Quick Plot

Just to make sure the result makes sense, here is a quick plot of the solution just derived -- the variance of the doubly truncated Binomial -- when $n = 12, p = \frac34, b= 12$, as the lower bound $a$ increases from 0 to 12. The thin red horizontal line is the unconditional variance of the Binomial parent (i.e. $np(1-p)$ ), and the the blue dots are the variance of the doubly truncated model, as the lower bound $a$ increases. As one would expect, the variance of the bounded model should be smaller than the unconstrained model. All looks fine.

Notes

1. The Prob and Var function used above are from the mathStatica package for Mathematica.

2. The variance calculation took about 10 hours to compute on a 2014 R2-D2 Mac Pro, though much of this would have been 'simplification' time.

3. Details on the Hypergeometric2F1 etc functions can be found here: http://reference.wolfram.com/language/ref/Hypergeometric2F1.html