Q.If $X\sim B(n,p)$ is a binomial random variable, prove that its mean is $np$.

Start from $E(X)=\sum_{r=0}^{n} r\cdot P(X=r)$, use the binomial probability formula, pull out a factor of $np$ using $r\binom{n}{r}=n\binom{n-1}{r-1}$, then collapse the remaining sum back into a full binomial expansion equal to 1.

By definition,

$E(X)=\displaystyle\sum_{r=0}^{n} r\binom{n}{r}p^rq^{n-r}$

The $r=0$ term vanishes, so

$E(X)=\displaystyle\sum_{r=1}^{n} r\cdot\dfrac{n!}{r!(n-r)!}p^rq^{n-r}=\displaystyle\sum_{r=1}^{n} n\cdot\dfrac{(n-1)!}{(r-1)!(n-r)!}p^rq^{n-r}$

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