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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!!

kjetil b halvorsen
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shafee
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  • 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 04:53
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    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 08:15
  • Your question has $n=4$ *exactly* so why the inequality $n\ge 5$? – kjetil b halvorsen Nov 28 '21 at 20:27

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Hints: You can fill in the details!

  • The distribution of the differences $X_1 - X_3$ and $X_2 - X_4$ do not depend on $\mu$, so we can assume $\mu=0$. This reduces to ordinary exponential distributions.

  • The distribution of the difference of two iid exponential distributions is a Laplace distribution.

  • The absolute value of a Laplace distributed random variable have an exponential distribution.

  • So, now we need the expectation of the exponential of an exponentially distributed random variable, see Distribution of the exponential of an exponentially distributed random variable?. This expectation will exist, under some condition on your parameter $\sigma$. I leave it for you to find the condition! (should be simple)

  • Assuming the condition from the former bullet point is fulfilled, you ask for the expectation of the difference of two iid variables. That is zero!

kjetil b halvorsen
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