construct an interval estimator (exponential)

I think I solved it but I don't have the answer in my book to check against.

Question.

Let be a random sample from an exponential distribution with density function

where is an unknown parameter. Define . Find the moment generating function of X. Show that is a pivotal function, and hence construct an interval estimator for with confidence coefficient 0.95. (Note. You may use the fact that the distribution with k degrees of freedom has moment generating function .)

Answer.

which, I think, accidentally means

Now we show that the distribution of does not depend on , so that it can be a pivotal function

Let

which is mgf for a chi-square distribution with 2n degrees of freedom.

So and the distribution of U does not depend on .

Now we construct an interval estimator for with confidence coefficient 0.95. Using the distribution of U,

I finally got the interval estimator .

Appreciate your feedback.