Answer to Question #99039 in Statistics and Probability for Aisha

Question #99039
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.
1
Expert's answer
2019-11-20T12:49:15-0500

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}"

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