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3.92. Find (a) the mode, (b) the median of a random variable X having density function f (x) and (c) compare with the mean. 3.93. Work Problem 3.100 if the density function is f (x) e 4x(1 0 x2) 0 x 1 otherwise e e 0
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3.94. Find (a) the median, (b) the mode for a random variable X defined by X e 2 1 prob. 1>3 prob. 2>3
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and (c) compare with the mean.
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3.95. Find (a) the median, (b) the mode of the set of numbers 1, 3, 2, 1, 5, 6, 3, 3, and (c) compare with the mean.
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Percentiles
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3.96. Find the (a) 25th, (b) 75th percentile values for the random variable having density function f (x) e 2(1 0 x) x 1 0 otherwise
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3.97. Find the (a) 10th, (b) 25th, (c) 75th, (d) 90th percentile values for the random variable having density function f (x) e c(x 0 x3) 0 x 1 otherwise
where c is an appropriate constant.
Other measures of dispersion
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.
CHAPTER 3 Mathematical Expectation
3.100. Find the mean deviation of the random variable X in each of the following cases.
(a) f(x)
x 0 otherwise
(b) f (x)
1 p(1 x2)
3.101. Obtain the probability that the random variable X differs from its mean by more than the semi-interquartile range in the case of (a) Problem 3.96, (b) Problem 3.100(a).
Skewness and kurtosis
3.102. Find the coefficient of (a) skewness, (b) kurtosis for the distribution of Problem 3.100(a). 3.103. If f (x) c Q1 0 u xu a R u xu u xu a a
where c is an appropriate constant, is the density function of X, find the coefficient of (a) skewness, (b) kurtosis. 3.104. Find the coefficient of (a) skewness, (b) kurtosis, for the distribution with density function f (x) e le 0
Miscellaneous problems
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 f (x) 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 f (x) e cx 0 0 x 2 otherwise e c(1 0 x) 0 x 1 otherwise
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. 3.109. Let X and Y have joint density function f (x, y) Find (a) E(X2 Y2), (b) E( !X2 Y2). e cxy 0 0 x 1, 0 otherwise y 1
3.110. Work Problem 3.109 if X and Y are independent identically distributed random variables having density function f (u) (2p) 1>2e u2>2, ` u `.
CHAPTER 3 Mathematical Expectation
3.111. Let X be a random variable having density function f (x) and let Y X2. Find (a) E(X), (b) E(Y), (c) E(XY). e2 0
1 x 1 otherwise
ANSWERS TO SUPPLEMENTARY PROBLEMS
3.43. (a) 1 (b) 7 (c) 6 3.45. (a) 1 (b) 2 (c) 1 3.48. (a) 1 (b) 1 (c) 1 > 4 3.50. (a) 7 > 10 (b) 6 > 5 (c) 19 > 10 (d) 5 > 6 3.52. (a) 2 > 3 (b) 2 > 3 (c) 4 > 3 (d) 4 > 9 3.54. (a) 7 > 12 (b) 7 > 6 (c) 5 > 12 (d) 5 > 3 (e) 7 > 4 (f) 2 > 3 3.55. (a) 5 > 2 (b) 55 > 12 (c) 1 > 4 (d) 1 > 4 3.56. (a) n (b) 2n 3.57. (a) 35 > 12 (b) !35>12 3.59. (a) 1 (b) 1 !5 (b) Var(X) = 3 > 80, sX 215>20 3.44. (a) 3 > 4 (b) 1 > 4 (c) 3 > 5 3.46. 10.5 3.47. 3
3.58. (a) 4 > 3 (b) !4>3 3.60. (a) Var(X) = 5, sX 3.61. (a) 4 (b) 2 3.63. (a) 2(et>2 3.64. (a) (1 3.65. (a) m1 3.66. (a) 1 > (1 3.67. m1 0, m2
