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    Uniform Distribution and Bernoulli Distribution.

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    1. Let -- be a random sample from a uniform distribution on --.
    a) Find the MLE --.
    b) Show that the distribution of -- does not depend on -- and use it to set 1 - alpha confidence interval for --.
    c) Show that the likelihood rate test statistic -- for testing --- against ---.

    2. Let -- be a random sample from a Bernoulli distribution with pmf --, zero elsewhere, where --
    a) We wish to test ---. Find -- and -- where -- is the likelihood ratio test statistics.
    b) Determine the Wald-type test and give, explicitly, the rejectiomn rule for a level alpha test for Part a).
    c) Determine the limiting distribution.

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    Solution Summary

    The solution illustrates the determination of MLE and likelihood ratio test statistic for a given problem. The response includes a Word and Excel attachment.