Mean of a Binomial Distribution

A random variable  
\[X\]
  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  
\[r=1\]
.
\[\begin{equation} \begin{aligned} \mu &= np \sum_{r=1}^n r \frac{(n-1)!}{r!(n-r)!} p^{r-1}r (1-p)^{n-r} \\ &= np \sum_{r=1}^n \frac{(n-1)!}{(r-1)!((n-1)-(r-1))!} p^{r-1}(1-p)^{(n-1)-(r-1)} \\ &= np \sum_{r=0}^{n-1} \frac{(n-1)!}{r!((n-1)-r)!} p^{r}r (1-p)^{(n-1)-r} \\ &= np \times 1 =np \end{aligned} \end{equation}\]

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