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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

Solution Preview

Solution.
<br>
<br>(a) Denote the mgf of W by M(t). We know that M(t)=E[e^(tW)]by definition. So,
<br> M(t)=E[e^(tW)]
<br> =e^(2t)k*log(3/2)+e^(3t)k*log(4/3)+
<br> e^(4t)k*log(5/4)+e^(5t)k*log(6/5)
<br>where k=1/log(3)
<br>
<br>(b) We know that
<br> E(W)=M'(0)
<br> =2k*log(3/2)+3k*log(4/3)+4k*log(5/4)+5k*log(6/5)
<br> E(W^2)=M"(0)
<br> ...

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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