Question

The moment generating fucntion of a random variable X is given by M(t)=(1-t/2)^-3.

Find the mean and variance of X

Answer 1

$$mean=\mu=E(X)=M'_t|_{t=0}$$

$$=(-3(-\dfrac{1}{2})(1-\dfrac{t}{2})^{-4})|_{t=0}=\dfrac{3}{2}$$

$$E(X^2)=M''_t|_{t=0}$$

$$=(-4(\dfrac{3}{2})(-\dfrac{1}{2})(1-\dfrac{t}{2})^{-5})|_{t=0}=3$$

$$Var(X)=E(X^2)-(E(X))^2=3-(\dfrac{3}{2})^2 =\dfrac{3}{4}$$