Q.If the mean and variance of a binomial distribution are 18 and 12 respectively, then $n = \ldots$ (A) $36$ (B) $54$ (C) $18$ (D) $27$

For $X\sim B(n,p)$, three summary numbers describe the whole distribution without listing every term of the p.m.f.: the mean $\mu=E(X)=np$ (the average number of successes over many repetitions of the $n$-trial experiment), the variance $\text{Var}(X)=npq$ (how spread out the successes are around that mean), and the standard deviation $SD(X)=\sigma_X=\sqrt{npq}$. These are stated as formulae without proof. They are used two ways: forward, when $n$ and $p$ are known, to compute $E(X)$ and $\text{Var}(X)$ directly; and backward, when two of $E( …