3. Consider X1, X2, . . . , Xn an i.i.d. sample of the exponential distribution with parameter ?. That is, the density is f(x|?) = ?e^-?x for x > 0 and 0 otherwise. (a) Show that X? is an m.v.u.e. of 1/?.
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Since X follows an exponential distribution with parameter 0, we have: F(x) = P(X ≤ x) = ∫0x f(t) dt = ∫0x 0e^-0t dt = 1 - e^0 = 1 for x > 0, and F(x) = 0 for x ≤ 0. Therefore, X has a degenerate distribution with all probability mass at x = 0. Show more…
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