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									  Homework Database

   Please show as many steps as
possible so you can get partial points
even if you don’t get the final answer.
              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).
              Homework 2
• Gelman book Exercise 2.1. θ is the
  probability that a coin will yield a “head”.
  Suppose the prior distribution on θ 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 θ on the same graph.(3pts)
               Homework 3
• Gelman book Exercise 2.2 (Predictive
  distributions) Two coins C1 and C2.
  Pr(heads|C1)=0.6, Pr(heads|C2)=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).
              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)
             Homework 5
• Find the conjugate prior for Poisson
  distribution.(2pts)
            Homework 6
• Gelman book 2.10 a) 2pts
          Homework 7-10
• Gelman book 2.21 (a,b,c,d) 1pt each
                   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.
a) give your posterior distribution for \theta (as a function
    of n) --2pts
b) If n=10, 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
    plot—1pt. Observe the relationship of the 3.
(Not required!) Re-Do b) c) with n=100
(Gelman book 2nd Ed. Ex 2.8)
               Homework 12
• Assume a non-
  informative improper
  prior on the standard
  deviation σ of a          p( )   1
  normal distribution is

• Prove that the
  corresponding prior
  density for σ2 is        p( )  
                              2        2

• 2pts (Gelman 2.19a)
                 Homework 13
• Pre-post debate polling. ABC news polled 639 voters
  before the 1988 presidential debate and different 639
  voters after. Let α1 be the proportion who favored Bush
  before the debate, and α2 after. Choose a
  noninformative prior. Plot a histogram of α2-α1. What is
  the posterior probability of a shift towards Bush (i.e. α2-
  α1>0)?

                     Bush       Dukakis    Other
          pre        294        307        38
          Post       288        332        19
                 Problem 21
The length of a light bulb has an exponential
   distribution with unknown rate θ.
a) Show that Gamma distribution is conjugate for
   θ given an independent and identically
   distributed sample of light bulb lifetimes.(2pts)
b) Suppose your prior for θ is a gamma
   distribution with coefficient of variation 0.5 (that
   is: sd/mean=0.5) 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)
                 Problem 22
• The football point spread problem, download the
  data from course homepage.
a)Using a noninformative prior on σ2,determine the
  posterior distribution for σ2 (2pts)
b)Suppose that we have prior belief we are 95%
  sure that σ 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.
              Problem 23
• Assume that the number of fatal accidents
  in each year are Poisson distributed with a
  constant unknown rate θ and an exposure
  proportional to the passenger miles flown
  that year(see table on next page). Set a
  prior distribution for θ and determine the
  posterior distribution based on the data.
  (4pts) Gelman 2.13. Table 2.2
       Fatal accidents Passenger miles
                        (100million)
1976   24               3863
1977   25               4300
1978   31               5027
1979   31               5481
1980   22               5814
1981   21               6033
1982   26               5877
1983   20               6223
1984   16               7433
1985   22               7107
                         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 σC, 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,logσc,logσt) (1 pt for writing down the joint
  prior for (μc,μt,σc2,σt2), 2 pt for deriving the posterior of μc, 1 pt for μt)
• What’s 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)
              Problem 32
• Knowing the posterior of σ2 is IG((n-1)/2,
  (n-1)S2/2 ) S2=(Sum of squares)/(n-1)
• Show that (n-1) S2/ σ2 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)
             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.
              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.

								
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