Expectation of the exponential function of Absolute Value of the Difference of Two Double parameter exponentially Distributed Random Variable

Suppose, $$X_1,\ldots, X_n$$ be iid having double parameter Exponential Distribution with common pdf $$f(x)= \dfrac{1}{\sigma} \exp\{ -(x-\mu)/\sigma \} I(x>\mu); \mu \in R, \sigma \in R^+ , n\ge5$$

Let, $$T=\exp(\lvert X_1 - X_3\rvert) - exp(\lvert X_2 - X_4\rvert)$$

Now I m trying to find $$E[T]$$. But I m really having no clue how to deal with this!!

Any help??

• The hard way is to compute the expectations. The easy way is to think about what the iid assumption tells you about the distributions of $|X_1-X_3|$ and $|X_2-X_4|:$ how do they differ? Use your answer to this (very easy) question to reason about how the expectations of their exponentials differ. – whuber Oct 19 '20 at 21:23
• Umm, I think since $X_is$ are iid , $\lvert X_1 - X_3\rvert$ and $\lvert X_2 - X_4\rvert$ would have same distribution. So is the answer is zero?? @whuber – shafee Oct 20 '20 at 4:53
• You also need to make sure that the expectation of each of the two terms at the end are finite for the argument to go through. I believe this requirement will place a bound on $\sigma$ (the answer is different otherwise). – Glen_b Oct 20 '20 at 8:15