Name: ___________________________________Bayesian Statistics, 22S:138Midterm 1, Fall 2008Show any computations that you carry out. Use the back of your exam paper if yourun out of space.1. Researchers choose a simple random sample of counties from amongst all thecounties in the U.S. The variable recorded for each county is whether the countycontains a city with population greater than 25,000. They wish to use these data toestimate the proportion of U.S. counties that have cities with over 25,000 people.(a) Which of the following parametric families is most likely to be appropriatefor the likelihood in their analysis?i. betaii. binomialiii. gammaiv. normalv. poisson(b) Justify your answer in a sentence or two.2. Researchers choose a simple random sample of counties from amongst all thecounties in the U.S. The variable reported for each of the selected counties is thenumber of miles of interstate highway that run through the county. Note that thisvariable does not have to take on only integer values.(a) Which of the following parametric families is most likely to be appropriatefor the likelihood in their analysis?i. betaii. binomialiii. gammaiv. normalv. poisson(b) Justify your answer in a sentence or two.13. A booth at the state fair lets people try to win a stuffed toy through a two-stepprocess. First the player must roll a fair die (6 faces, each equally likely to comeup) to determine which game he plays next. If the die comes up as a one or a two,the player will play a game called the Duck Shoot. If the die comes up 3, 4, 5, or6, the player will play a game called the Hen Hunt. The chance of winning thestuffed toy in the Duck Shoot is 1/3. The chance of winning the toy in the HenHunt is 1/10.(a) What is the probability that a player will get the Hen Hunt for step two?(b) What is the overall probability that a player will win the stuffed toy?(c) Your nephew comes up to you carrying the stuffed toy he won at the booth.What is the probability that he played the Hen Hunt, given that he won thetoy?24. A biologist believes that lifetimes in days of fruitflies follow an exponential distri-bution. The density for a single observation y drawn from an exponential distri-bution isp(y; λ) = λe−λy, 0 < y < ∞For data, the biologist will observe 50 randomly selected fruitflies and record howmany days they live.(a) Write the likelihood of λ if there are 50 data values, represe nted symbolicallyas y1, . . . , y50.(b) What parametric family is the conjugate prior for the exponential likelihood?(Just name the family.)(c) The biologist states that a prior distribution with m ean 3 days and vari-ance 12 would reflect his previous knowledge about fruitfly lifetimes. Whatparameters should he specify in his prior? (numeric answer; show your work)(d) Based on the likelihood and the prior that you have specified, write an ex-pression to which the posterior density p(λ|y) is proportional.(e) Is this a density that you recognize? If so, name it and give the numericvalues of the
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