Question

Given a random variable with binomial distribution use moment generating technique to determine the mean

Answer 1

The moment generating function for binomial distribution:

$f_x(t)=\Epsilon_x(e^{tx})=\sum_{k=0}^{n}e^{tk}C_n^kp^k(1-p)^{n-k}=\\ =\sum_{k=0}^{n}C_n^k(pe^t)^k(1-p)^{n-k}=(pe^t+(1-p))^n\\ f_x^{(1)}(t)=n(pe^t+(1-p))^{n-1}pe^t$

Mean is the first central moment and is calculated by the next formula:

$\mu=f_x^{(1)}(0)=n(pe^0+(1-p))^{n-1}pe^0=np$