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3.64. (a) Find the moment generating function of a random variable X having density function (b) Use the generating function of (a) to find the first four moments about the origin. 3.65. Find the first four moments about the mean in (a) Problem 3.43, (b) Problem 3.44. 3.66. (a) Find the moment generating function of a random variable having density function and (b) determine the first four moments about the origin. 3.67. In Problem 3.66 find the first four moments about the mean.

3.68. Let X have density function . Find the kth moment about (a) the origin, (b) the mean. 3.69. If M(t) is the moment generating function of the random variable X, prove that the 3rd and 4th moments

3.70. Find the characteristic function of the random variable . 3.71. Find the characteristic function of a random variable X that has density function 3.72. Find the characteristic function of a random variable with density function 3.73. Let be independent random variables (k 1, 2, . . . , n). Prove that the characteristic function of the random variable is 3.74. Prove that as the characteristic function of Problem 3.73 approaches (Hint: Take the logarithm of the characteristic function and use L'Hospital's rule.)

3.76. Work Problem 3.75 if the joint density function is . 3.77. Find (a) Var(X), (b) Var(Y), (c) X, (d) Y, (e) XY, (f) , for the random variables of Problem 2.56. 3.78. Work Problem 3.77 for the random variables of Problem 2.94. 3.79. Find (a) the covariance, (b) the correlation coefficient of two random variables X and Y if E(X) 2, E(Y) 3, E(XY) 10, E(X2) 9, E(Y2) 16. 3.80. The correlation coefficient of two random variables X and Y is while their variances are 3 and 5. Find the covariance

3.81. Let X and Y have joint density function Find the conditional expectation of (a) Y given X, (b) X given Y. 3.82. Work Problem 3.81 if 3.83. Let X and Y have the joint probability function given in Table 2-9, page 71. Find the conditional expectation of (a) Y given X, (b) X given Y. 3.84. Find the conditional variance of (a) Y given X, (b) X given Y for the distribution of Problem 3.81. 3.85. Work Problem 3.84 for the distribution of Problem 3.82. 3.86. Work Problem 3.84 for the distribution of Problem 2.94.

3.87. A random variable X has mean 3 and variance 2. Use Chebyshev's inequality to obtain an upper bound for (a) P( X 3 2), (b) P( X 3 1). 3.88. Prove Chebyshev's inequality for a discrete variable X. (Hint: See Problem 3.30.) 3.89. A random variable X has the density function f(x) (a) Find P( u X mu 2). 3.98. Find (a) the semi-interquartile range, (b) the mean deviation for the random variable of Problem 3.96. 3.99. Work Problem 3.98 for the random variable of Problem 3.97.

3.105. Let X be a random variable that can take on the values 2, 1, and 3 with respective probabilities 1 3, 1 6, and 1 2. Find (a) the mean, (b) the variance, (c) the moment generating function, (d) the characteristic function, (e) the third moment about the mean. 3.106. Work Problem 3.105 if X has density function where c is an appropriate constant. 3.107. Three dice, assumed fair, are tossed successively. Find (a) the mean, (b) the variance of the sum. 3.108. Let X be a random variable having density function where c is an appropriate constant. Find (a) the mean, (b) the variance, (c) the moment generating function, (d) the characteristic function, (e) the coefficient of skewness, (f) the coefficient of kurtosis.

CHAPTER 3 Mathematical Expectation 101 3.64. (a) Find the moment generating function of a random variable X having density function f(x) = Jx/2 05x52 lo otherwise (b) Use the generating function of (a) to find the first four moments about the origin. 3.65. Find the first four moments about the mean in (a) Problem 3.43, (b) Problem 3.44. 3.66. (a) Find the moment generating function of a random variable having density function f(x) = to x 20 otherwise and (b) determine the first four moments about the origin. 3.67. In Problem 3.66 find the first four moments about the mean. 3.68. Let X have density function f(x) = [1/(b - a) asx=b lo otherwise . Find the kth moment about (a) the origin, (b) the mean. 3.69. If M() is the moment generating function of the random variable X, prove that the 3rd and 4th moments about the mean are given by 43 = M"(0) - 3M"(O)'(0) + 2[A(O)] (4 = M"(O) - 4M"(O'(0) + 6MOM(OF - 3[M(O)]Law of large numbers 3.90. Show that the (weak) law of large numbers can be stated as np ( si - 1