Homework Database

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Homework Database

• Please show as many steps as possible so you can
  get partial points even if you dont get the
  final answer.

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Homework 1

• Suppose your prior distribution for the
  proportion of Californians who support the death
  penalty has mean 0.6 and standard deviation 0.3.
  Use a beta distribution for this prior. A random
  sample of 1000 californians is taken, 65 support
  the death penalty. Plot the prior and posterior
  on the same plot. (3pts) (Gelman book 2.9).

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Homework 2

• Gelman book Exercise 2.1. ? is the probability
  that a coin will yield a head. Suppose the
  prior distribution on ? is Beta(4,4). The coin is
  spun 10 times, heads appeared fewer than 3
  times. Calculate your exact posterior density.
  Using a graphic software, plot prior and
  posterior of ? on the same graph.(3pts)

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Homework 3

• Gelman book Exercise 2.2 (Predictive
  distributions) Two coins C1 and C2.
  Pr(headsC1)0.6, Pr(headsC2)0.4. Now choose
  one coin randomly and spin it repeatedly. The
  first two spins are tails, what is the
  expectation of the number of additional spins
  until a head shows up? (3pts)
• If you knew which coin was chosen, the two spins
  are independent of each other. Show that not
  knowing which one, the two spins are not
  independent(1pt).

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Homework 4

• We observed y female births out of a total of n
  births. Prove that the posterior predictive
  probability of the next two births are not
  independent, i.e. the posterior predictive
  probability of the next two births are both
  female is not equal to the posterior predictive
  probability of the next birth being female
  -squared. (2pts)

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Homework 5

• Find the conjugate prior for Poisson
  distribution.(2pts)

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Homework 6

• Gelman book 2.10 a) 2pts

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Homework 7-10

• Gelman book 2.21 (a,b,c,d) 1pt each

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Homework 11

• A random sample of n students is drawn from a
  large population. Their average weight is 150
  pounds. Assume that the weight in the population
  are normally distributed with mean \theta and
  standard deviation 20. Suppose your prior for
  \theta is normal with mean 180 and standard
  deviation 40.
• give your posterior distribution for \theta (as a
  function of n) --2pts
• b) If n10, give 95 posterior interval for
  \theta.1pt
• c) Using a graphic software, plot the prior, the
  likelihood (as a function of \theta) and
  posterior in the same plot1pt. Observe the
  relationship of the 3.
• (Not required!) Re-Do b) c) with n100
• (Gelman book 2nd Ed. Ex 2.8)

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Homework 12

• Assume a non-informative improper prior on the
  standard deviation s of a normal distribution is
• Prove that the corresponding prior density for s2
  is
• 2pts (Gelman 2.19a)

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Homework 13

• Pre-post debate polling. ABC news polled 639
  voters before the 1988 presidential debate and
  different 639 voters after. Let a1 be the
  proportion who favored Bush before the debate,
  and a2 after. Choose a noninformative prior. Plot
  a histogram of a2-a1. What is the posterior
  probability of a shift towards Bush (i.e.
  a2-a1gt0)?

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Problem 21

• The length of a light bulb has an exponential
  distribution with unknown rate ?.
• Show that Gamma distribution is conjugate for ?
  given an independent and identically distributed
  sample of light bulb lifetimes.(2pts)
• Suppose your prior for ? is a gamma distribution
  with coefficient of variation 0.5 (that is
  sd/mean0.5) A random sample of light bulbs is to
  be tested to measure their lifetime. If the
  coefficient of variation is to be reduced to 0.1,
  how many light bulbs should be tested? (2pts)
  Gelman book 2.21 a) c)

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Problem 22

• The football point spread problem, download the
  data from course homepage.
• a)Using a noninformative prior on s2,determine
  the posterior distribution for s2 (2pts)
• b)Suppose that we have prior belief we are 95
  sure that s falls between 3 and 20 points. Find
  an approximate conjugate prior that corresponds
  to this belief (can do this by trial and error on
  the computer) 2pts
• Gelman 2.23.

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Problem 23

• Assume that the number of fatal accidents in each
  year are Poisson distributed with a constant
  unknown rate ? and an exposure proportional to
  the passenger miles flown that year(see table on
  next page). Set a prior distribution for ? and
  determine the posterior distribution based on the
  data. (4pts) Gelman 2.13. Table 2.2

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Problem 31

• Gelman 3.3 An experiment was performed on the
  effects of magnetic fields on the flow of calcium
  out of chicken brains. Measurements on an
  unexposed group of 32 chickens had a sample mean
  of 1.013 and sample standard deviation of 0.24.
  Measurements on exposed group of 36 chickens had
  sample mean of 1.173 and sample sd of 0.20.
• A) assuming that the measurements in the control
  (unexposed) group were from a normal distribution
  with mean µC and standard deviation sC, what is
  the posterior distribution of µC? Similarly, what
  is the posterior distribution of the treatment
  group µt? Assume a uniform prior on
  (µc,µt,logsc,logst) (1 pt for writing down the
  joint prior for (µc,µt,sc2,st2), 2 pt for
  deriving the posterior of µc, 1 pt for µt)
• Whats the posterior distribution of µt-µc? You
  can obtain independent posterior samples of µt
  and µc then plot the histogram of the difference.
  (2pts) Obtain an approximate 95 posterior
  interval for µt-µc (1pts. Hint use the
  quantile() function)

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Problem 32

• Knowing the posterior of s2 is IG((n-1)/2,
  (n-1)S2/2 ) S2(Sum of squares)/(n-1)
• Show that (n-1) S2/ s2 is a chi-square with n-1
  degrees of freedom
• (2pts. Hint InverseChisquare is a different
  parameter form of InverseGamma. The inverse of an
  inverseChi-square is a Chi-square)

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Problem 41

• Gelman book 2nd Ed. 5.9 a) 3pts. b) 1pt c) 1pt.
  Hint choice of noninformative priors are on
  pages 134 and 136.

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Problem 42

• 5.13. Formulate the full model, clarify what each
  parameter corresponds to (2pts)
• a) 2pt
• c) 2pts
• Describe what each graph is and the corresponding
  algebra and code, if any.