If X1 and X2 are iid ~ exp(m) and Y=X1-X2, The moment generating function of y would be
$E(e^{ty})=E(e^{t({x1-x2})})=\frac{E(e^{tx1})}{E(e^{tx2})}= \frac{\frac{m}{m-t}}{\frac{m}{m-t}}=1$
Why is this wrong?
2026-03-29 15:54:54.1774799694
mgf of Y=X1-X2; in which X1 and X2 are from the same distribution
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It's wrong because in general the expectation of a ratio is not the ratio of the expectations. In other words you cannot write $$E\left(\frac XY\right)=\frac{E(X)}{E(Y)}.$$ If $Y$ and $X/Y$ were independent, then this would be legal, but it's not true in your case.