(The gamma as a mixture of exponentials) Gleser (1989) shows that, in certain cases, the gamma distribution can be written as a scale mixture of exponentials, an identity suggested by different analyses of the same data. Let f{x) be a gamma(r, λ) pdf.
(a) Show that if r

where

(Make a change of variable from v to u, where u = x/u - x/λ.)
(b) Show that pλ(v) is a pdf, for r

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m(v) = m(v) dv = 1. 읊 log (f(z)) > 0 but 읊 log (Joo(e-z/v/v)gx (v)dv) < 0, a contradiction.)