# Let denote the proportion of registered voters in a large city who are in favor of

## Question:

Let θ denote the proportion of registered voters in a large city who are in favor of a certain proposition. Suppose that the value of θ is unknown, and two statisticians A and B assign to θ the following different prior p.d.f.’s ξA(θ) and ξB(θ), respectively:
ξA(θ) = 2θ for 0 < θ <1,
ξB(θ) = 4θ3 for 0 < θ <1.
In a random sample of 1000 registered voters from the city, it is found that 710 are in favor of the proposition.
a. Find the posterior distribution that each statistician assigns to θ.
b. Find the Bayes estimate for each statistician based on the squared error loss function.
c. Show that after the opinions of the 1000 registered voters in the random sample had been obtained, the Bayes estimates for the two statisticians could not possibly differ by more than 0.002, regardless of the number in the sample who were in favor of the proposition.
Distribution
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