If a random variable U has a gamma distribution with parameters > 0 and > 0, then Y = eU

Chapter 6, Problem 6.112

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If a random variable U has a gamma distribution with parameters > 0 and > 0, then Y = eU [equivalently, U = ln(Y )] is said to have a log-gamma distribution. The log-gamma distribution is used by actuaries as part of an important model for the distribution of insurance claims. Let U and Y be as stated. a Show that the density function for Y is f (y) = 1 () y(1+)/ (ln y)1, y > 1, 0, elsewhere. b If < 1, show that E(Y ) = (1 ). [See the hint for part (c).] c If 0 and > 0, and that the moment-generating function of a gamma-distributed random variable only exists if t < 1; see Example 4.13.]