Question

given that \( x \) is a random variable from a binomial distribution with parameter \( (n, p) \), Using the m.g.f technique show that the expected value and variance of \( x \) is \( n p \) and \( n p q \) respectively

Answer 1

For Binomial distribution

$f(x)=\binom{n}{x}p^xq^{n-x}\\ M(t)=E(e^{tx})=e^{tx}\sum_{x=0}^n\binom{n}{x}p^xq^{n-x}\\ =\sum_{x=0}^n\binom{n}{x}(pe^t)^xq^{n-x}\\ =(pe^t+q)^n\\ M'(t)=npe^t(pe^t+q)^{n-1}\\ M'(0)=E(x)=np(p+q)^{n-1}=np\\ M''(t)=npe^t(pe^t+q)^{n-2}(n-1)pe^t+npe^t(pe^t+q)^{n-1} \\ M''(0)=np(p+q)^{n-2}(n-1)p+np(p+q)^{n-1}\\ M''(0)=np(n-1)p+np=n(n-1)p^2+np\\ var(x)=M''(0)-(M''(0))^2\\ =n(n-1)p^2+np-(np)^2\\ =np-np^2=np(n-p)=npq$