Expand the moment–generating function of the gamma distribution as a binomial series, and read off the values of µ'1, µ'2, µ'3, and µ'4.
Answer to relevant QuestionsUse the results of Exercise 6.13 to find α3 and α4 for the gamma distribution. A random variable X has a Pareto distribution if and only if its probability density is given by Where α > 0. Show that µ'r exists only if r < α. If a random variable X has a uniform density with the parameters a and β, find its distribution function. Use the Maclaurin’s series expansion of the moment-generating function of the standard normal distribution to show that (a) µr = 0 when r is odd; (b) µr = r! 2r / 2r/2(r/2)! when r is even. If X and Y have a bivariate normal distribution and U = X + Y and V = X – Y, Find an expression for the correlation coefficient of U and V.
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