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CHAPTER 11 Bayesian Methods
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11.135. The number of accidents per month on a particular stretch of a highway is known to follow the Poisson distribution with parameter l. A total of 24 accidents occurred on that stretch during the past 10 months. What are the chances that there would be fewer than four accidents there next month Assume Jeffreys prior for l: p(l) 1> !l, l 0. 11.136. Suppose that all 10 out of 10 Bernoulli trials were successes. What are the chances that all five out of five future Bernoulli trials would be successes Assume a uniform prior density for the probability of success. 11.137. A sample of size 20 from a normal population with unknown mean u and variance 4 yields a sample mean of 37.5. The prior distribution of u is normal with mean 30 and variance 3. Suppose that an independent observation from the same population is subsequently made. Find the predictive probability that this observation would not exceed 37.5.
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11.102. (a) subjective; (b) classical; (c) frequency; (d) insufficient information: an equally convincing case could be made for this being a classical, frequency, or subjective probability; (e) subjective 11.103. 2 > 7 11.104. The posterior distribution is given in Table 11-14. Table 11-14 l p(l u x) 11.105. p(uu x) 1 4, n ux 3(1 0.5 0.42 u)n x, 0 1.0 0.41 u 1 1.5 0.17
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11.106. The posterior density is beta with parameters 3 and 9. 11.107. The posterior density of u is beta with a 11.108. The posterior density of u is beta with a 33 and b 68 and b 19. 34. nb) 5>6 < 0.83;
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11.109. (a) The posterior density is gamma with parameters nx a 14 2 16 and b>(1 # (b) the posterior mean 80>6 < 13.33; (c) the posterior variance 400>36 < 11.11 11.110. The posterior density is gamma with parameters nx # 11.111. (a) The posterior mean of u is (c) the precision is about 1.78 16 16 x 9 # a 42 4 46 and b>(1
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2>21 < 0.10.
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16x # ; (b) the precision is roughly 4.34; 25
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11.112. The posterior density is normal with mean 1.98 and variance 0.0043. 11.113. The posterior density of l is gamma with parameters 35.5 and 0.1 (see Problem 11.25). 11.114. The posterior density is inverse gamma with a 2 and b 16.875 (see Problem 11.26).
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CHAPTER 11 Bayesian Methods
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11.115. The posterior density is beta with parameters 21 and 81 (see Example 11.12). 11.116. The posterior density is gamma with parameters 11 and 0.02. 11.117. The posterior density is inverse gamma with parameters 3 and 7.75. 11.118. (a) 0.11; (b) 0.11 a a
11.119. (a) E(u ux) (a b B(x B(x nr)(b nx # nx 1)(a # 3, n 2, n x x
11.120. (a)
nr b nx #
; (b)
b(a n) 1 ; (b) E u x 1 nbx u # nr) b nx) #
1 b(n ax 2
nbx # a 1) b ^ n 2 a 1
ax n 3
11.123. n
2) 1)
n(x 2)(n x 1) (n 4)(n 3)
11.121.
11.124. [0.25, 0.75]
11.125. [0.13, 0.30]
11.126. [0.10, 0.28]
11.127. (a) [0.078, 0.196]; (b) [5.10, 12.82]
11.128. The posterior probability of the null hypothesis is 0.02. Since this is less than 0.05, we reject the null hypothesis. 11.129. (a) The posterior probability of the null hypothesis is 0.04. Since this is not less than 0.025, we cannot reject the null hypothesis. (b) The posterior probability of the null hypothesis is 0.026. Since this is not less than 0.025, we cannot reject the null hypothesis. (c) The posterior probability of the null hypothesis is 0.015. Since this is less than 0.025, we reject the null hypothesis. 11.130. The posterior probability of the null hypothesis is 0.03. Since this is less than 0.05, we reject the null hypothesis. 11.131. The posterior odds ratio is 0.66/0.34 reject the null hypothesis. 1.94 and the prior odds ratio is 6>4 1.5. BF 1.29. We cannot
11.132. Reject the null hypothesis since the posterior probability of the null hypothesis is 0.033 while the prior probability of the null hypothesis is 0.216. y 9 9 6 10 5 y ,y 11 11 1 0.240 2 0.288
11.133. f *( y)
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