Question: Let X~Exp(θ) and Y~Exp(θ) be independent. Find the moment generating function of X-Y and identify its distribution.

Okay so the mgf of an exponential random variable is θ/(θ-t), so I got the mgf of X-Y to be (θ^2)/((θ^2)-(t^2)). Firstly, is this correct?

Whether or not it is I don't understand how to identify the distribution of X-Y given the mgf. I've looked at mgfs of several standard distributions and nothing seems to match up. Most distributions have two parameters and we only have one here (θ) so I don't understand what would happen if X-Y had more than one parameter.

I feel as though I am lacking some understanding of what is actually going on, can anyone help?

  • Yes, you are correct. There is a constraint $t < \theta$. I believe what you are looking for is the double exponential distribution. – eev2 Dec 1 '15 at 13:06
up vote 2 down vote accepted

Your calculation is correct. You can write that MGF also as $$\frac{1}{1-t^2/\theta^2},$$ which looking at the Wikipedia article on MGFs, appears to be the MGF of a $\mathrm{Laplace}(0,1/\theta^2)$ distribution.

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.