Find a confidence interval using the particular statistics function

Question

This problem was a part of my assignment, which is over now, but I would like to know the right answer to it. Assume that $X_1,...,X_n$ are non-negative i.i.d. r.v.s with p.d.f. $$f(x | θ) = 2 θ e^{−2θx}$$

1. Find the distribution of the statistics $T := X1 + · · · + Xn$.
2. Find the distribution of the statistics $4θT$.
3. Using the above, find a $100(1 − α)\%$ confidence interval for $θ$.

I know how to solve all of the above, but how is point 2 supposed to help find the confidence interval?

Answer 1

It' s a chi square distribution

$4\theta T\sim\chi_{(2n)}^2$

This is useful to calculate the Confidence interval using chi square-table.

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As an example, fix $n=10$ and $\alpha\%=10\%$ and try to calculate the proposed CI (with equiprobable tails)

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Solution

Using the chi-square table we immediately get

$$10.85<4\theta\Sigma_iX_i<31.41$$

That is

$$\frac{10.82}{4\Sigma_iX_i}<\theta<\frac{31.41}{4\Sigma_iX_i}$$