For any random variable Z with MGF: M_Z (t)=(1/3 exp(t)+ 2/3)^5 show the distribution of Z and obtain E(Z) and Var(Z) !

QUESTION:

For any random variable Z with MGF:

$M_{Z}(t)=(\frac{1}{3}exp(t)+\frac{2}{3})^5$

show the distribution of Z and obtain $E(Z)$ and $Var(Z)$ !

ANSWER:

Random variable Z has MGF: $M_{Z}(t)=(\frac{1}{3}exp(t)+\frac{2}{3})^5$ , which is the MGF of the Binomial distribution with parameters $n=5$ and $p=1/3$ or $Z\sim Binomial(5,\frac{1}{3})$.

$E(Z)$:

With MGF so for $E(Z)=M^{'}_{Z}(0)$ 

then

$Var(Z)$:

$Var(Z)=M^{''}_{Z}(0)-(M^{'}_{Z}(0))^{2}$

for $M^{''}_{Z}(t)$:

so,

You can find $E(Z)$ and $Var(Z)$ using an easy method as we know for $E(Z)=np$ and $Var(Z)=np(1-p)$ in a Binomial distribution with parameters n and p, then:

$E(Z)=np=5 . \frac{1}{3}= \frac{5}{3}$

$Var(Z)=np(1-p)=5 . \frac{1}{3}(1-\frac{1}{3})=5 . \frac{1}{3} . \frac{2}{3}=\frac{10}{9}$