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    Approximate Distribution: Central Limits and Delta

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    2) Let X1, X2,... be i.i.d. random variables all having a Gamma distribution with parameters α and 3. Let n = (1/n) ∑i=1nXi. Define h(x) = √(2x).
    1) Find the 0.5 quantile of the approximate distribution of n using the central limit theorem. You must show all steps.
    2) Find the approximate distribution of h( n) using to the delta method. You must show all steps.

    © BrainMass Inc. brainmass.com October 7, 2022, 6:35 pm ad1c9bdddf
    https://brainmass.com/statistics/central-limit-theorem/approximate-distribution-central-limits-delta-169416

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    2) Let X1, X2,... be i.i.d. random variables all having a Gamma distribution with parameters α and 3. Let n = (1/n) ∑i=1nXi. Define h(x) = √(2x).
    1) Find the 0.5 quantile of the approximate distribution of n using the central limit theorem. You must show all steps.

    Solution. As X1, X2,... are i.i.d. random variables all having a Gamma distribution with parameters α and 3, we have

    , i=1,2,...,n. ..............(1)

    NOTE: (See on Page 224 of a textbook: A First Course in Probability Theory, 5th edition, Sheldon Ross) If X is a Gamma distribution with parameters t and , then

    ,

    Let t= and =3. We get the above in (1).

    So, by the linearity of expectation, we have

    ................(2)
    Similarly, by a property of variance, we have

    .........(3)
    So, by the central limit theorem, when n is large, roughly follows a normal distribution with mean and variance . Hence,

    when n is large.

    So, the 0.5 quantile of the approximate distribution of n is .

    2) Find the approximate distribution of h( n) using to the delta method. You must show all steps.

    Solution. By 1), we know that roughly follows a normal distribution with mean and variance . To get the approximate distribution of h( n), we denote its distribution function as F(x), namely,

    So,
    i) when , we have

    ii) when x>0, we have

    ....................(*(

    As for large n, roughly follows a normal distribution with mean and variance . Hence,

    So, by (*), we have

    where

    is the distribution function of the standard normal distribution N(0,1).

    So, we get the approximate distribution of h( n) as follows.

    where .

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

    © BrainMass Inc. brainmass.com October 7, 2022, 6:35 pm ad1c9bdddf>
    https://brainmass.com/statistics/central-limit-theorem/approximate-distribution-central-limits-delta-169416

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