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11.88. In Problem 11.21, find the distribution of the mean of a future sample of size m.
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0. Normalizing this gamma density, we get
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11.89. The number of accidents per month on a particular stretch of a highway is known to follow the Poisson distribution with mean l. A total of 24 accidents occurred on that stretch during the past 10 months. What are the chances that there would be more than 3 accidents there next month Assume Jeffreys prior for l: p(l) 1> !l, l 0.
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The predictive distribution of the number of accidents Y during the next month may be obtained from Problem 11.88 with n 10, nx 24, m 1: # 1024 f *( y) y! 24 f *(1)
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yi! nx #
nx #
1 (n 2
my #
1 1)nx my 2
,y #
0 1 2 m , m , m , c.
The probability we need is 1 0.236.
[ f *(0)
f *(2)
1 1124 2
1 y 2
1 2
0, 1, 2, c.
f *(3)]
[0.097
0.201]
CHAPTER 11 Bayesian Methods
11.90. In Problem 11.65, what are the chances that the number of bad reactions next year would not exceed 1
We need the predictive distribution for one future observation. We have the posterior in Problem 11.65 as gamma with parameters 20.5 and 0.1. Combining this with the probability function of Y, we get f (y; l) f (y ul)p(lu x) e lly 1020.5l19.5e y! (20.5)
0, 1, 2, . . . . ; l
The marginal probability function for Y, obtained by integrating out l, is
f *( y)
1020.5ly 19.5e y! (20.5)
1020.5 (y 20.5) y! (20.5)11y 20.5
The probabilities corresponding to y-values 0 through 7 are given in Table 11-10. The probability that the number of bad reactions would be 0 or 1 is 0.4058. Table 11-10 y f *( y) 0 0.1417 1 0.2641 2 0.2581 3 0.1760 4 0.0940 5 0.0419 6 0.0162 7 0.0056
11.91. In Theorem 11-4, suppose that another, independent sample of size 1 is drawn from the exponential population. (a) Determine its predictive distribution. (b) Estimate the result of the future observation using the predictive mean.
(a) Denote the future observation by Y. We then have the following joint distribution of Y and the posterior density of u. ue f ( y; u) f ( yuu)p(uu x)
uy(1
nbx)n # bn
a un a 1e u A 1 n xB
nbx)n # bn a
a un ae u A 1 n x yB
Integrating out u,
*( y)
nbx)n a un # bn a (n nbx)n # nbx #
a b(n
a e u A 1 n x yB
a) a)
(1 (1
nbx)n abn a 1 (n # nbx by)n a 1bn #
a (n
1) a)
(1 (1
by)n
y(1 (b) The mean of this predictive distribution is 3 (1
nbx)n ab(n # nbx by)n #
1 b(n
nbx # . a 1)
11.92. In Problem 11.29, find the predictive density and predictive mean of a future observation.
*( y)
y r
1 1 B(a
1 nr, b nr B(a
11.93. A couple has two children and they are both autistic. Find the probability that their next child will also be autistic assuming that the incidence of autism is independent from child to child and has the same probability u. Assume that the prior distribution of u is (a) uniform, (b) beta with parameters a 2, b 3.
(a) Applying Theorem 11-11 with n B(3 distribution of Y is f *( y) child will be autistic is 3 > 4. 2, x y, 2 B(3, 1) 2, m 1, and a y) (2 y)!(1 8 b y)! 1, we see that the predictive ,y 0, 1. The probability that the next
1 B(a 1
nx #
ua nr) 3
nr r 1(1
u)b r)
n x y nr r 1 du,
r, r
1, . . .
r, b nx y nr # nr, b nx nr) #
r, r
1, c
CHAPTER 11 Bayesian Methods
(b) Applying Theorem 11-11 with n B(4 distribution of Y is f *( y) child will be autistic is 4>7. 2, x y, 4 B(4, 3) 2, m 1, and a y) (3 y)!(3 84 2, b y)! ,y 3, we see that the predictive 0, 1. The probability that the next
11.94. A random 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 5. Suppose that an independent observation is subsequently made from the same population. Find (a) the predictive probability that this observation would not exceed 37.5 and (b) the equal tail area 95% predictive interval for the observation. From Theorem 11-12, the predictive density is normal with mean 37.21 and standard deviation 2.05.
(a) Equals the area to the left of 0.14 under the predictive density: 0.56 (b) 37.21 (1.96 2.05) [33.19, 41.23]
11.95. All 10 tosses of a coin resulted in heads. Assume that the prior density for the probability for heads is p(u) 6u5, 0 u 1 and find (a) the predictive distribution of the number of heads in four future tosses, (b) the predictive mean, and (c) the predictive mode.
(a) Note that the prior density is beta with parameters a 6 and b 1. From (19), with m 10, n 4, 4 B(16 y, 5 y) a 6, b 1, and x 10, we get f *( y) , y 0, 1, 2, 3, 4. The numerical B(16, 1) y values are shown in Table 11-11. Table 11-11 1 0.0033 2 0.0281 y f *( y) 0 0.0002 3 0.1684 4 0.8000
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