Question: Let X i , . . . , X n be an independent and identically distributed sequence of Exponential(1) random variables, where n 3. Find

Let X_i, . . . , X_n be an independent and identically distributed sequence of Exponential(1) random variables, where n 3. Find the conditional probability density function for the maximum M = X_(n) given that the second smallest X₍₂₎ = 1. Use this to verify that, in the case that n = 4,

E(M X₍₂₎ = 1) = 2e2 2e.

Hint: Start by using the "PDF method" to find the density for X₍₂₎and the joint density for (M, X₍₂₎).

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