THE UNIVERSITY OF IOWA 22S:175 HW Set 2 Due at the beginningof class on Monday, January 31, 2011 Write legibly. Show your steps. Explain your reasoning. Simplify your answers. 1. (An extension of #4 in HW Set 1) Let X be a random variable for which the moment generating function, MX(t) = E[etX], exists. Let and 2 denote its mean and variance, respectively. For j = 3, 4, 5, ... , let j = E[(X – μ)j] denote its j-th central moment. Expand ln[MX(t)] as a Maclaurin seriesof t, lnMX(t) = 0jjtj/ j! . Determine the first five coefficients, 0, 1, … , 4, in terms of μ, 2, μ3and μ4. Remarks: (i) is the Greek letter kappa. (ii) The function ln MX(t) is called K(t) in Exercise 79 on page 95 of Ross. (iii) The following result is very useful in actuarial science: A random variable X is normalif and only if j= 0 for all j ≥3. (iv) A Maclaurin series is a special case of the Taylor series. 2. Let X and Y be two independent exponential random variables with mean Xand Y, respectively. Evaluate the conditional expectation E[X | X < Y].
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