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Math Help - Poisson Distribution Proof and Exercise

  1. #1
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    Poisson Distribution Proof and Exercise

    Two related questions here....

    Consider a random sample X1,X2, . . . ,Xn from a Poisson(μ) distribution. Using the log–likelihood function, show that the maximum likelihood estimator
    for μ is given by ˆμ = ¯X . (That is meant to say "moo hat = X bar"

    Suppose the number of buses arriving each rush–hour (8am – 9am) at a bus–
    stop in Chicago is thought to follow a Poisson(μ) distribution. In five consecutive daily rush–hours, the numbers of buses arriving at the bus–stop are:

    22, 14, 23, 15, 16.

    Find the maximum likelihood estimate for μ. What assumption have you made
    about the five measurements?
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  2. #2
    MHF Contributor matheagle's Avatar
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    The likelihood function is

    {e^{-n\mu}\mu^{\sum x_i}\over \prod (x_i !)}

    so take the log of that, the log likelihood function is then

    -n\mu +\sum x_i\lg \mu -\lg(\prod (x_i !))

    Differentiate this with respect MOO and set it equal to 0.

    The solution/point estimator is the sample mean.

    You should take the second derivative to prove it's a max, or use the first derivative test.
    Not many people do that, but you should.

    As for traffic flow in chicago, there are plenty of jokes I can make.
    I called a few years ago complaining about how the red lights are not in phase.
    The guy in charge is a former student from my school.
    The light one block from my apartment is still is out of phase.
    Last edited by matheagle; October 20th 2009 at 01:05 AM.
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