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3.62. If a random variable X is such that E[(X
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Moments and moment generating functions
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3.63. Find (a) the moment generating function of the random variable X and (b) the first four moments about the origin. e 1>2 1>2 prob. 1>2 prob. 1>2
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CHAPTER 3 Mathematical Expectation
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3.64. (a) Find the moment generating function of a random variable X having density function f (x) e x>2 0 0 x 2 otherwise
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(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) e e 0
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and (b) determine the first four moments about the origin. 3.67. In Problem 3.66 find the first four moments about the mean. e 1>(b 0 a) a x b . Find the kth moment about (a) the origin, otherwise
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3.68. Let X have density function f (x) (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 about the mean are given by m3 m4 M-(0) M(iv)(0) 3Ms(0)Mr(0) 4M-(0)Mr(0) 2[Mr(0)]3 6Ms(0)[Mr(0)]2 3[Mr(0)]4
Characteristic functions
3.70. Find the characteristic function of the random variable X e a b prob. p prob. q 1 . p
3.71. Find the characteristic function of a random variable X that has density function f (x) e 1>2a 0 a u xu otherwise
3.72. Find the characteristic function of a random variable with density function f (x) e e x>2 0 0 x 2 otherwise 1, 2, . . . , n). Prove that the characteristic
3.73. Let Xk
1 prob. 1>2 be independent random variables (k 1 prob. 1>2 function of the random variable X1 is [cos (v> !n)]n. X2 c 2n
3.74. Prove that as n S ` the characteristic function of Problem 3.73 approaches e the characteristic function and use L Hospital s rule.)
v2>2.
(Hint: Take the logarithm of
CHAPTER 3 Mathematical Expectation
Covariance and correlation coefficient
3.75. Let X and Y be random variables having joint density function f (x, y) e x 0 y 0 x 1, 0 otherwise y 1
Find (a) Var(X ), (b) Var(Y ), (c) sX, (d) sY, (e) sXY, (f) r.
e e 0
(x y)
3.76. Work Problem 3.75 if the joint density function is f (x, y)
x 0, y otherwise
3.77. Find (a) Var(X), (b) Var(Y ), (c) sX, (d) sY, (e) sXY, (f) r, 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 ) E(XY) 10, E(X2) 9, E(Y2) 16. 3.80. The correlation coefficient of two random variables X and Y is covariance.
2, E(Y )
while their variances are 3 and 5. Find the
Conditional expectation, variance, and moments
3.81. Let X and Y have joint density function f (x, y) e x 0 y 1, 0 0 x otherwise y 1
Find the conditional expectation of (a) Y given X, (b) X given Y. e 2e 0
(x 2y)
3.82. Work Problem 3.81 if f (x, y)
x 0, y otherwise
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.
Chebyshev s inequality
3.87. A random variable X has mean 3 and variance 2. Use Chebyshev s inequality to obtain an upper bound for (a) P( u X 3 u 2), (b) P( u X 3 u 1). 3.88. Prove Chebyshev s inequality for a discrete variable X. (Hint: See Problem 3.30.)
1 3.89. A random variable X has the density function f (x) 2 e |x|, ` Chebyshev s inequality to obtain an upper bound on P(u X m u
x `. (a) Find P( u X m u 2). (b) Use 2) and compare with the result in (a).
CHAPTER 3 Mathematical Expectation
Law of large numbers
3.90. Show that the (weak) law of large numbers can be stated as Sn lim P 2 n e 1 0 m2 P 1
and interpret.
3.91. Let Xk (k = 1, . . . , n) be n independent random variables such that Xk prob. p prob. q 1 p
(a) If we interpret Xk to be the number of heads on the kth toss of a coin, what interpretation can be given to Sn X1 c Xn (b) Show that the law of large numbers in this case reduces to
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