- #1

Locoism

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## Homework Statement

Assume Y

_{1}, ... , Y

_{n}is a sample of size n from a gamma distributed population with α = 2 and unknown β

a) Use the method of moment generating functions to show that

[itex] U = 2 \sum_1^n Y_i / β [/itex]

is a pivotal quantity and has a chi-square distribution with 4n d.f.

b) Use U to derive a 95% confidence interval for β

c) If a sample of size n = 5 yields y-bar = 5.39, use the result from b to give a 95% confidence interval for β

## Homework Equations

2Y

_{i}/β has a chi square with 4 d.f.

## The Attempt at a Solution

I've done part a) easily, just take E[e

^{tU}] = (1-2t)

^{-2n}so U has a chi-square distribution with 4n d.f.

Par b is giving me more trouble, since I don't know how to start it. Usually, I would look at a table to find numbers (a,b) so that

P(a < U < b) = 0.95

by taking first P(U < a) = 0.025 and P(b < U) = 0.025 from the tables.

But since I don't know the degrees of freedom, how do I know which values to choose?