Question

Let X have MGF MX (t) =1/8(1+ e^t )
a) Find E(X) using MX(t)
b) Find the variance of X using MX(t).
c) Find the moment generating function of Y=2X-1.

Answer 1

We can write

$$E[X^k]={d^k \over dt^k}M_X(t)\big|_{t=0}$$

a)

$$M_X(t)={1+e^t\over 8}$$

$$E[X]={d \over dt}M_X(t)\big|_{t=0}=({e^t\over 8})\big|_{t=0}={1\over 8}$$

$$E[X]={1\over 8}$$

b)

$$E[X^2]={d^2 \over dt^2}M_X(t)\big|_{t=0}=({e^t\over 8})\big|_{t=0}={1\over 8}$$

$$Var[X]=E[X^2]-(E[X])^2$$

$$Var[X]={1\over 8}-({1\over 8})^2={7\over 64}$$

$$Var[X]={7\over 64}$$

c)

If $Y=aX+b,$ then

$$M_Y(t)=e^{tb}M_X(at)$$

$$Y=2X-1$$

$$M_Y(t)=e^{-t}M_X(2t)=e^{-t}\cdot{1+e^{2t}\over 8}={e^{-t}+e^{t}\over 8}=$$

$$={\cosh{t}\over 4}$$

$$M_Y(t)={\cosh{t}\over 4}$$