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.

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