University of TorontoMississauga

STA312H5F- Fall, 2008

Test 1

September26, 2008

Last Name:______

First Name:______

Student Number:______

Instructions:

  • Time: 50 minutes.
  • No aids allowed except a nonprogrammable calculator.
  • Answers that are algebraic expressions should be simplified. Series and integrals should be evaluated whenever required. Numerical answers need not be expressed in decimal form.
  • If you do not understand a question, or are having some other difficulty, do not hesitate to ask your instructor for clarification.
  • There are 7 pages including this page. Please check that you are not missing any page.
  • Show all your work and answer in the space provided, in ink. Pencil may be used, but then remarks will not be allowed. Use back of pages for rough work.
  • Total point: 40.

Good luck!!

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Suppose you toss coin until you observe the first head. Your friend who gave you that coin told you that there is a much higher chance of observing a head than a tail.

1) (5 points) What is the statistical model for this problem? (Hint: start by defining the

appropriate random variable, identify it’s distribution and the parameter of interest)

Define: X = the number of tosses until we observe the first head.

The parameter of interest here is θ = Pr(head), that is, θ is the probability of

observing a head in any given toss.

We have that

The sampling model is then the set: whereis probability mass

function of a Geometric(θ) random variable given by

2) (5 points) Suppose that a Beta distribution on θ is an appropriate prior for this

problem. In the next page you will find two plots: the Beta(2,5) distribution and the Beta(5,2) distribution. Which one of these two distributions is an appropriate choice of a prior for this problem in light of the information your friend gave you about the coin? Explain!

Your friend believes that there is a much higher chance of observing a head than a tail, this means that θ is more likely to be higher than 0.5. The Beta(5, 2) distribution place most of its mass on θ values that are higher than 0.5, that is, the area under the prior density curve above 0.5 is much bigger than the corresponding area under the Bets(2, 5) density curve. Therefore, the Beta(5, 2) is the appropriate choice of a prior for this problem.

3) (5 points) Find the inverse normalizing constant using the prior you chose in part (2).

(Hint: the inverse normalizing constant is simply the prior predictive distribution of the variable you defined in part 1).

where x = 1, 2, ….

4) (5 points) Find the posterior distribution of θ using the prior you chose in part (2) and identify it.

which is the Beta(6, x+1) distribution.

5) (5 points) Suppose you had to toss the coin 10 times until you observed the first head.

What is the posterior mean and posterior variance? Use your answer to part (4).

If we had to toss the coin 10 times before we observed the first head, it means that we observe x = 10. The posterior distribution is then the Beta(6, 11) distribution.

The posterior mean is .

The posterior variance is

END!

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