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Approximate Distribution: Central Limits and Delta

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)
...

Solution Summary

This solution calculates the approximate distribution of a Gamma distribution using the central limit theorem and the delta method.

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