# Is the mean of the left-truncated binomial distribution convex in p?

The expectation of the binomial distribution of successes in $$G$$ trials, left-truncated at $$R$$, with success probability $$p$$, is

$$E[X|p] = \frac{\sum_{l=R}^Gl\phi(l)}{\sum_{l=R}^G\phi(l)}$$

where

$$\phi(l) = \binom{G}{l}p^l(1-p)^{G-l}.$$

Is this convex in $$p$$? It looks as if it is.

• Start with the definition of a convex function, and then see if this definition applies here. Have you tried this already? yesterday
• I have written it out, but proving the relevant inequality looks hard! $aE[X|p] +(1-a)E[X|q] > E[X|ap+(1-a)q]$... it's not easy to see how this will simplify 19 hours ago