Explore BrainMass

Explore BrainMass

    Moment Generating Function

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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

    © BrainMass Inc. brainmass.com March 4, 2021, 5:55 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/moment-generating-function-18139

    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

    $2.49

    ADVERTISEMENT