Moment Generating Function
2007-09-20 5:28 am
Given the moment-generating function Mx(t)= exp (3t + 8t ^2), find the moment generating function of the random variable Z=1/4 (X-3), and use it to determine the mean and the variance of Z.
回答 (1)
2007-09-20 9:01 am
✔ 最佳答案
Mz(t) = int[exp (zt) f(x) dx]= int[exp(1/4(x-3)t) f(x) dx]
= exp(-3/4t) int[exp(1/4 tx) f(x)dx]
= exp(-3/4t) Mx(1/4t)
= exp(-3/4t) exp(3/4 t + 1/2 t^2)
= exp(1/2 t^2)
Mz'(t) = t exp(1/2 t^2), Mz''(t) = exp(1/2 t^2) + t^2 exp(1/2 t^2)
mean(Z) = Mz'(0) = 0
Var(Z) = Mz''(0) - (Mz'(0))^2 = 1
參考: me
收錄日期: 2021-04-23 20:04:36
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https://hk.answers.yahoo.com/question/index?qid=20070919000051KK03844