Solved – Mean and variance of log-binomial distribution

We can use an entirely analogous technique to the one typically used to calculate the moments of a lognormal.

In particular, note that if $newcommand{E}{mathbb E}X sim mathrm{Bin}(n,p)$ and $Y = e^X$, then $Y^k = e^{k X}$. But, $E e^{kX} = m_X(k)$ where $m_X(t)$ is the moment-generating function of $X$ evaluated at $t$.

Hence, $$ E e^{k X} = m_X(k) = (1 – p + p e^k)^n >, $$ and so $$ E Y = m_X(1) = (1 + (e-1)p)^n $$ and $$ mathrm{Var}(Y) = m_X(2) – (m_X(1))^2 = (1+(e^2-1)p)^n – (1+(e-1)p)^{2n} >. $$

Other moments of $Y$ can be computed in a similar fashion.