variance of doubly truncated binomial 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
 A: 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


*

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

*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.

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