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    Probability: Moment Generating Functions and Poisson Process

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    1.) Let X be a discrete random variable with probability mass function
    Pr {X=k} = c(1+ k^2) for k= -2, -1, 0, 1, 2.
    a) Determine c.
    b) Determine Pr {X <= 0}
    c) Determine the mean of X
    d) Why is the previous answer fairly obvious?
    e) Determine the variance of X
    f) Compute Pr {X=2 | X >= 0}
    g) Determine the moment generating function of X

    2.) Let Y be a Poisson random variable with parameter 1.5
    a) Compute Pr {Y=0}
    b) Compute Pr {Y <= 1}
    c) Compute Pr {Y=0 |Y <= 1}
    d) What is E[Y]?
    e) Determine the variance of Y
    f) Determine the moment generating function

    3) Suppose the random variable S has moment generating function of
    (q + p *(e^t))^n where q= 1- p, 0 < p < 1 and n is a positive integer. Find the mean and
    variance of Y.

    © BrainMass Inc. brainmass.com March 4, 2021, 5:45 pm ad1c9bdddf
    https://brainmass.com/math/probability/probability-moment-generating-functions-poisson-process-9351

    Solution Preview

    1.a.)
    because total probability = 1
    => sum(P[k] for k = -2,-1,0,1,2) = 1
    => P(-1) + P(-2) + P(0) + P(1) + P(2) = 1
    => c.(1+(-2)^2) + c.(1+(-1)^2) + c.(1+(0)^2) + c.(1+1^2) + c.(1+2^2)=1
    => c*{(1+4) + (1+1) + (1+0) + (1+1) + (1+4)} = 1
    => c*(5+2+1+2+5) = 1
    => c = 1/15 --Answer

    b.)
    P(X<=0) == P(X =-2) + P(X=-1) + P(X = 0)
    => P(X<=0) = (1/15)*{(1+(-2)^2) + (1+(-1)^2 + (1+0^2)}
    => P(X<=0) = (1/15)*{5+2+1) = 8/15 --Answer

    c.)
    mean of X = <X>
    <X> = (for Xi =-2,-1,0,1,2)sum{Xi * P(Xi)}
    => <X> = {(-2)*P(-2) + (-1)*P(-1) + 0*P(0) + 1*P(1) + 2*P(2)}
    => <X> = {-2*(1+(-2)^2) - (1+(-1)^2) + 0 + (1+1^2) + ...

    Solution Summary

    Probabilities are determined and moment generating functions are calculated. The mean functions are calculated.

    $2.49

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