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p(uu x) ~ unr(1 u)n(x r) ua 1(1 ters a nr and b nx nr. # u)b
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11.30. The interarrival time of customers at a bank is exponentially distributed with mean 1>u, where u has a gamma distribution with parameters a 1 and b 0.2. Twelve customers were observed over a period of time and were found to have an average interarrival time of 6 minutes. Find the posterior distribution of u.
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Applying Theorem 11-4 with a 1 and b 0.2, n 11 (12 customers 1 11 interarrival times), x # we see that the posterior density is gamma with parameters 12 and 0.014. 6,
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11.31. In the previous problem, suppose that a second, independent sample of 10 customers was observed and was found to have an average interarrival time of 6.5 minutes. Find the posterior distribution of u.
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Since the gamma family is conjugate for the exponential distribution, this problem can be done in two ways: (i) by starting with the prior gamma distribution with parameters 1 and 0.2 and applying Theorem 11-4 with n 11 9 20 and x ((11 6) (9 6.5))>20 < 6.225 or (ii) by starting with the prior gamma # distribution with parameters 12 and 0.014 and applying Theorem 11-4 with n 9. Both ways lead to the result that the posterior density is gamma with parameters 21 and 0.0077.
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11.32. The following density is known as a Rayleigh density: f (x) uxe (ux2>2), x 0. It is a special case of the Weibull density (see page 118), with b 2 and a u. Show that the gamma family is conjugate with respect to the Rayleigh distribution. Specifically, show that if X has a Rayleigh density and u has a gamma prior density with parameters a and b, then the posterior density of u given a random sample x (x1, x2, c, xn) of observations from X is also a gamma.
p(uu x) ~ f (x u u) p(u) ~ une parameters a n and 2b 2 b a x2
ua xi2>2
ua 1e
~ u(a
n) 1 e uQ
1 x2 >2R b a i , i
0. This is a gamma density with
11.33. Show that the inverse gamma family (see Problem 11.99) is conjugate with respect to the normal distribution with known mean but unknown variance u.
CHAPTER 11 Bayesian Methods
Assume that the mean of the normal density is 0. We have n 1 1 2 f (xu u) exp e a xi f n>2un>2 2u i 1 (2p) p(u) The posterior density is given by p(uu x) f (xu u) p(u) ~ u
n> 2 e
1 2u
a 1e b>u
a xi
a 1e b>u ~u(
axi n 1 i R Qb a) 1e 2 , 2 u
n This is also an inverse gamma, with parameters 2
a xi a and b
11.34. A random sample of n observations is taken from the exponential density with mean u: f (xu u) (1>u) exp5 x>u6, x 0. Assume that u has an inverse gamma prior distribution (see Problem 11.99) and show that its posterior distribution is also in the inverse gamma family.
f (xu u) p(u) (1>u)n exp e bau
a 1e b>u
a xi >u f , x
a xi
The posterior density, given by p(u u x) ~ f (x u u) p(u) ~ u ne is inverse gamma with parameters n a and b a xi.
a 1e b>u
(n a) 1e
1 Qb aixiR, u
11.35. In the previous problem, suppose that a second sample of m observations from the same population yields the observations y1, y2, c, ym. Find the posterior density incorporating the result from both samples.
Since the inverse gamma family is conjugate with respect to the exponential distribution, we can update the posterior parameters obtained in Problem 11.34 to m density is thus inverse gamma with parameters m n (n a) and Qb a xi
a xi R
a yj. The posterior
a and b
a yj.
11.36. A random sample of n observations is taken from the gamma density:
f (x u u) xa 1e x>u , x 0. Assume that u has an inverse gamma prior distribution with parameters g and b ua (a) and show that its posterior distribution is also in the inverse gamma family. f (xu u) ~ (1>ua)n exp e p(u) bgu
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