Description
Problem 1
Suppose that 6 observations are taken at random from a uniform distribution on the interval (θ−1/2, θ+1/2),
with θ unknown, and that their values are 11.0, 11.5, 11.7, 11.1, 11.4 and 10.9. Suppose that the prior
distribution of θ is a uniform distribution on the interval (10,20). Determine the posterior distribution of θ.
Problem 2
Suppose that the time in minutes required to serve a customer at a certain facility has an exponential distribution for which the value of the parameter θ is unknown and that the prior distribution of θ is a gamma
distribution for which the mean is 0.2 and the standard deviation is 1. If the average time required to serve
a random sample of 20 customers is observed to be 3.8 minutes, what is the posterior distribution of θ?
Problem 3
Show that the family of beta distributions is a conjugate family of prior distributions for samples from
a negative binomial distribution with a known value of the parameter r and an unknown value of the
parameter p, with 0 < p < 1.
Problem 4
Sample survey: Suppose we are going to sample 100 individuals from a county (of size much larger than
100) and ask each sampled person whether they support policy Z or not. Let Yi = 1 if person i in the sample
supports the policy, and Yi = 0 otherwise.
1. Assume Y1, . . . , Y100 are, conditional on θ, i.i.d. binary random variables with expectation θ. Write
down the joint distribution of Pr(Y1 = y1, . . . , Y100 = y100 | θ) in a compact form. Also write down the
form of Pr(PYi = y | θ).
2. For the moment, suppose you believed that θ ∈ {0.0, 0.1, …, 0.9, 1.0}. Given that the results of the
survey were P100
i=1 Yi = 57, compute Pr(P100
i=1 Yi = 57) for each of these 11 values of θ and plot these
probabilities as a function of θ.
3. Now suppose you originally had no prior information to believe one of these θ-values over another,
and so Pr(θ = 0.0) = P r(θ = 0.1) = . . . = P r(θ = 0.9) = P r(θ = 1.0). Use Bayes’ rule to compute
p(θ |
P100
i=1 Yi = 57) for each θ-value. Make a plot of this posterior distribution as a function of θ.
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4. Now suppose you allow θ to be any value in the interval [0, 1]. Using the uniform prior density for θ,
so that p(θ) = 1, plot the posterior density p(θ) × Pr(P100
i=1 Yi = 57 | θ) as a function of θ.
5. As discussed in this chapter, the posterior distribution of is beta(1+ 57, 1 + 100-57). Plot the posterior
density as a function of θ. Discuss the relationships among all of the plots you have made for this
exercise.
Problem 5
Sensitivity analysis: It is sometimes useful to express the parameters a and b in a beta distribution in terms
of θ0 = a/(a + b) and n0 = a + b, so that a = θ0n0 and b = (1 − θ0)n0. Reconsidering the sample survey data
in Problem 4, for each combination of θ0 ∈ {0.1, 0.2, . . . , 0.9} and n0 ∈ {1, 2, 8, 16, 32} find the corresponding
a, b values and compute Pr(θ > 0.5 |
PYi = 57) using a beta(a, b) prior distribution for θ. Display the
results with a contour plot, and discuss how the plot could be used to explain to someone whether or not
they should believe that θ > 0.5, based on the data that P100
i=1 Yi = 57.
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