STAT330 Lecture Notes - Boundary Value Problem, Random Variable, Partition Coefficient
9 Feb 2014
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Exercise 2. 2. 6 (page 8) and properties of gamma function (page 9) If x is a random variable with p. f. f (x) = show that. Solution 1: you can answer this question by using the following power series result about log(1 y), i. e. , log(1 y) = (cid:88) yx x x=1. The above result can be obtained from page 26, 2. 10. 9 (logarithmic series) or from wikipedia: http://en. wikipedia. org/wiki/taylor series. 1 log p log(1 (1 p)) = 1. Solution 2: if you do not remember the logarithmic series and you still know the summation of geometric series, then you may consider the following method. (cid:88) x=1 x x=1 x d{ (1 p)x}/dp. Note that the boundary condition g(1) = 0 and g(cid:48)(p) = 1/p implies that g(p) = log p. (cid:88) x=1 f (x) = g(p) = 1. Note: derivative helps us to cancel the x on the denominator, which simpli es the summation. This is a commonly used trick in the power series.
