Moment Generating Function
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
(c) Determine and sketch the distribution function of W
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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 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
(c) Determine and sketch the distribution function of W
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