Find the moment generating function of the discrete random variable?

Question

X which has probability function
p(x)= 2(1/3)^x for x= 1, 2, 3,...
and use it to determine the value of u= E[X].
(Hint: Recall that Sum(from x=0 to infinity) ar^x= a/(1-r) for IrI< 1.)

Answer 1

m_X(t) = E[e^(tX)]

= 2 [ (1/3) e^t + (1/3)² e^(2t) + (1/3)³ e^(3t) + ... ]
= 2 [ (1/3) e^t + (1/3)² (e^(t))² + (1/3)³ (e^(t))³ + ... ]
= 2 [ (e^t)/3 + ((e^t)/3)² + ((e^t)/3)³ + ...]

To use the hint, we need to factor out (e^t)/3, so

m_X(t) = 2 (e^t)/3 [1 + (e^t)/3 + ((e^t)/3)² + ((e^t)/3)³ + ...]

= [2 (e^t)/3] / [1 - (e^t)/3] (for (e^t)/3 < 1)

which may be simplified (by multiplying numerator and denominator by 3) to

(2 e^t) / [3 - (e^t)]

Now, differentiate this with respect to t and then evaluate at t=0. This will give you µ. You should get µ = 3/2 .