175HWSet2-2011 - THE UNIVERSITY OF IOWA 22S:175 HW Set 2...

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THE UNIVERSITY OF IOWA 22S:175 HW Set 2 Due at the beginning of 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, M X (t) = E[e tX ], 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[M X (t)] as a Maclaurin series of t, l n M X (t) = 0 j j t j / j! . Determine the first five coefficients, 0 , 1 , … , 4 , in terms of μ, 2 , μ 3 and μ 4 . Remarks : (i) is the Greek letter kappa . (ii) The function ln M X (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 normal if 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 X and Y , respectively. Evaluate the conditional expectation E[X | X < Y].
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