The Cauchy distribution

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4.105. Suppose that X is Cauchy distributed according to (29), page 114, with a (b) P(X 2 12). 2. Find (a) P(X 2),

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4.106. Prove that if X1 and X2 are independent and have the same Cauchy distribution, then their arithmetic mean also has this distribution. 4.107. Let X1 and X2 be independent and normally distributed with mean 0 and variance 1. Prove that Y Cauchy distributed. X1 > X2 is

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The gamma distribution

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4.108. A random variable X is gamma distributed with 3, 2. Find (a) P(X 1), (b) P(l X 2).

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The chi-square distribution

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4.109. For a chi-square distribution with 12 degrees of freedom, find the value of x2 such that (a) the area to the right c of x2 is 0.05, (b) the area to the left of x2 is 0.99, (c) the area to the right of x2 is 0.025. c c c 4.110. Find the values of x2 for which the area of the right-hand tail of the x2 distribution is 0.05, if the number of degrees of freedom v is equal to (a) 8, (b) 19, (c) 28, (d) 40. 4.111. Work Problem 4.110 if the area of the right-hand tail is 0.01.

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CHAPTER 4 Special Probability Distributions

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4.112. (a) Find x2 and x2 such that the area under the x2 distribution corresponding to v 20 between x2 and x2 is 1 2 1 2 0.95, assuming equal areas to the right of x2 and left of x2. (b) Show that if the assumption of equal areas in 2 1 part (a) is not made, the values x2 and x2 are not unique. 1 2 4.113. If the variable U is chi-square distributed with v 7, find x2 and x2 such that (a) P(U 1 2 (b) P(U x2) 0.50, (c) P(x2 U x2) 0.90. 1 1 2 4.114. Find (a) x2 and (b) x2 for v 0.05 0.95 4.115. Find (a) x2 and (b) x2 for v 0.025 0.975 150. 250. x2) 2 0.025,

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Student s t distribution

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4.116. For a Student s distribution with 15 degrees of freedom, find the value of t1 such that (a) the area to the right of t1 is 0.01, (b) the area to the left of t1 is 0.95, (c) the area to the right of t1 is 0.10, (d) the combined area to the right of t1 and to the left of t1 is 0.01, (e) the area between t1 and t1 is 0.95. 4.117. Find the values of t for which the area of the right-hand tail of the t distribution is 0.01, if the number of degrees of freedom v is equal to (a) 4, (b) 12, (c) 25, (d) 60, (e) 150. 4.118. Find the values of t1 for Student s distribution that satisfy each of the following conditions: (a) the area between t1 and t1 is 0.90 and v 25, (b) the area to the left of t1 is 0.025 and v 20, (c) the combined area to the right of t1 and left of t1 is 0.01 and v 5, (d) the area to the right of t1 is 0.55 and v 16. 4.119. If a variable U has a Student s distribution with v 10, find the constant c such that (a) P(U (b) P( c U c) 0.98, (c) P(U c) 0.20, (d) P(U c) 0.90. c) 0.05,

The F distribution

4.120. Evaluate each of the following: (a) F0.95,15,12; (b) F0.99,120,60; (c) F0.99,60,24; (d) F0.01,30,12; (e) F0.05,9,20; (f) F0.01,8,8.

ANSWERS TO SUPPLEMENTARY PROBLEMS

4.61. (a) 1> 64 (b) 3 > 32 (c) 15 > 64 (d) 5 > 16 (e) 15 > 64 (f) 3 > 32 (g) 1> 64 4.62. (a) 57 > 64 (b) 21> 32 4.64. (a) 250 (b) 25 (c) 500 4.67. 193 > 512 4.63. (a) 1> 4 (b) 5 > 16 (c) 11> 16 (d) 5 > 8 4.65. (a) 17> 162 (b) 1> 324 4.66. 64 > 243

4.68. (a) 32 > 243 (b) 192 > 243 (c) 40 > 243 (d) 242 > 243 0.1127 (d) 2.927 6pq) 3n2p2q2 4.72. (a) 1.5, 1.6 (b) 72, 90