\[X\]
follows a binomial distribution \[X \im B(n,p)\]
. The expected valee of \[X\]
is the mean of the binomial distribution \[B(n,p)\]
is\[\begin{equation} \begin{aligned} E(X)= \mu &= \sum_{r=0}^n r P(X=r) \\ &= \sum_{r=0}^n r{}^nC_r p^r (1-p)^{n-r} \\ & = \sum_{r=0}^n r \frac{n!}{r!(n-r)!} p^r (1-p)^{n-r} \\ &= np \sum_{r=0}^n r \frac{(n-1)!}{r!(n-r)!} p^{r-1}r (1-p)^{n-r} \end{aligned} \end{equation}\]
The
\[r=0\]
term makes no contribution to the summation, so we can start the summation from \[r=1\]
.\[\begin{equation} \begin{aligned} \mu &= np \sum_{r=1}^n r \frac{(n-1)!}{r!(n-r)!} p^{r-1}r (1-p)^{n-r} \\ &= np \sum_{r=1}^n \frac{(n-1)!}{(r-1)!((n-1)-(r-1))!} p^{r-1}(1-p)^{(n-1)-(r-1)} \\ &= np \sum_{r=0}^{n-1} \frac{(n-1)!}{r!((n-1)-r)!} p^{r}r (1-p)^{(n-1)-r} \\ &= np \times 1 =np \end{aligned} \end{equation}\]