Why p$\frac{\partial }{\partial P} $ is the mean of binomial distribution function?

Question

Why p$\frac{\partial }{\partial P}[ {p+q}]^n $ is the mean of binomial distribution function? I know the mean should be $\sum np(n) $ but why p$\frac{\partial }{\partial P} [{p+q}]^n$ of the binomial distribution is the mean. And i dont see that to be used in Poisson or Gaussian distribution. So why is this the mean ?

Answer 1

Using the binomial expansion,

$$ \sum_{i=1}^n i\binom{n}{i}p^iq^{n-i}=p\times\sum_{i=1}^n\binom{n}{i}\frac{\partial}{\partial p}\left(p^iq^{n-i}\right)=p\times \frac{\partial}{\partial p}(p+q)^n $$