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If X1, X2 are iid exp(1), Let Y=X1-X2, why moment generating function of Y is 1/(1-t^2)

user399659
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  • Welcome to Math.SE. Often a terse problem statement of this kind is taken by many Readers to indicate no thought has been given by the Original Poster to digesting the problem themselves. This is a reasonable Question in the sense that it can be resolved by reasoned mathematical argument, but it would be an improvement if you added context. For example, where does this problem arise? What approach did you consider, and where did you run into difficulty? Is the problem with definitions or mechanics (computation)? Why do you like this problem? Not everything, but something can be said. – hardmath Dec 16 '16 at 18:31

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Note \begin{aligned} E[\exp(tY)]&=E[\exp(tX_1-tX_2)]=E\left[\exp(tX_1)\exp(-tX_2)\right]\\ &=E[\exp(tX_1)]E[\exp(-tX_2)]. \end{aligned} Because $X_1\sim\exp(1)$, we have $E[\exp(tX_1)]=\frac{1}{1-t}$ for $t<1$. It remains to note \begin{aligned} E[\exp(-tX_2)]&=E[\exp((-t)X_2)]=\frac{1}{1-(-t)}\quad\text{for}\quad(-t)<1. \end{aligned} Thus, $$ \text{MGF of }Y:\quad E[\exp(tY)]=\frac{1}{1-t}\frac{1}{1+t}=\frac{1}{1-t^2},\quad -1<t<1. $$

yurnero
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