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Suppose we have the following set of independent and identically distributed exponential random variables: [tex]\gamma_1,\,\gamma_2,\ldots ,\,\gamma_N[/tex]. If we arrange them in ascending order we get the following order statistics: [tex]\gamma^{(1)}\leq\gamma^{(2)}\leq\cdots\leq\gamma^{(N)}[/tex].

I need to find the moment generating function (MGF) of the highest order statistics, i.e.: [tex]\mathcal{M}_{\gamma^{(N)}}(s)=E_{\gamma^{(N)}}[\text{e}^{s\,\gamma}][/tex] in terms of the MGFs of the exponential RVs. Is there any way to connect these MGFs?

Thanks in advance

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# The MGF of Order Statistics

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