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Moment Generating Function

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The discrete rv W has the pmf

pw(W) = klog[(w+1)/w] for w = 2,3,4,5;
0 otherwise

k = 1/ log(3)

(a) deduce the mgf of W

(b) Calculate E[W] and var[W] using the mgf

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Solution Summary

The discrete rv W has the pmf

pw(W) = klog[(w+1)/w] for w = 2,3,4,5;
0 otherwise

k = 1/ log(3)

(a) deduce the mgf of W

(b) Calculate E[W] and var[W] using the mgf

Solution Preview

Solution.

(a) Denote the mgf of W by M(t). We know that M(t)=E[e^(tW)]by definition. So,
M(t)=E[e^(tW)]
=e^(2t)k*log(3/2)+e^(3t)k*log(4/3)+
e^(4t)k*log(5/4)+e^(5t)k*log(6/5)
where k=1/log(3)

(b) We know that
E(W)=M'(0)
=2k*log(3/2)+3k*log(4/3)+4k*log(5/4)+5k*log(6/5)
E(W^2)=M"(0)
...

Solution provided by:

Changping Wang, MA

Education

BSc , Wuhan Univ. China
MA, Shandong Univ.

Recent Feedback

"Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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"Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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