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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.

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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 .

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