Moment-generating function of discrete random variable

👨🏻‍🏫 學術台數學

Apricot

2007-12-05 6:21 am

Find the moment-generating function of discrete random variable X that has the following probability distribution

f(x)=2(1/3)^x for x=1, 2, 3...

I know the answer is

Mx(t)=2e^t/3-e^t

Please show me the step.

更新1:

對, 這條的要點是利用Moment-generating function的定義: Mx(t)=E(e^tx) 和 Geometric Series: a+ar+ar^2+ar^3+..=a/(1-r) for |r|

更新2:

|r| smaller than 1

回答 (1)

myisland8132

2007-12-05 7:09 am

✔ 最佳答案

Mx(t)
=E(e^tx)
=(e^t)(2/3)+(e^2t)2(1/3)^2+(e^3t)2(1/3)^3+...
=2[(e^t/3)+(e^t/3)^2+(e^t/3)^3+...]
=2(e^t/3)/(1-(e^t/3))
=2e^t/3-e^t

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