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    Convergence in Probability

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    A. If g is a real-valued function that is continuous at a, then g(a^) converges in probability to g(a).
    b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0 < y <1, 0 elsewhere. Show that converges in probability to some constant and find the constant.

    © BrainMass Inc. brainmass.com December 24, 2021, 7:28 pm ad1c9bdddf
    https://brainmass.com/statistics/probability/convergence-probability-mathematical-statistics-192937

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    1. Proofs
    a. Prove Theorem 9.2(d): If g is a real-valued function that is continuous at , then converges in probability to .
    b. Let Y1, Y2, ...Yn be independent random variables, each with probability density function for 0  y 1, 0 elsewhere. Show that converges in probability to some constant and find the constant.
    Solution
    a) Since is consistent for , given and , however small, we can find an such that
    , whenever . (1)
    Again, since g is a real-valued function that is continuous at , given , we can choose such that

    Thus
    [From (1)]
    That is, , whenever .
    Hence converges in probability to .

    [Note that means and hence or equivalently ]

    b) Since the p.d.f. of Y is given by
    for 0  y 1, 0 elsewhere
    we have
    =
    =
    and hence =
    Let .
    Then

    and

    Since and as , converges in probability to the constant = 0.75

    [Note that if is a sequence of estimators of such that and as , then is a consistent estimator of ]

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

    © BrainMass Inc. brainmass.com December 24, 2021, 7:28 pm ad1c9bdddf>
    https://brainmass.com/statistics/probability/convergence-probability-mathematical-statistics-192937

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