Hello,
It looks OK
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.