Reference no: EM131109070

Honors Exam - Spring 2009 Probability

(I) Exercises-

1. Three people A, B and C play a game in which they throw coins, one after the other. A starts, then B, then C, then A again etcetera. The person who throws heads first wins the games. Construct an appropriate sample space for this game, and find the probability that A wins the game.

2. (a) Suppose that A and B are independent events. Show that A^c and B are independent.

(b) Prove Bonferroni's inequality P(A ∩ B) ≥ P(A) + P(B) - 1.

(c) Prove the general version of Bonferroni's inequality P(∩ⁿ_i=1A_i) ≥ i=1Σⁿ P(A_i) - (n - 1).

3. An urn contains n white and m black balls, where n and m are positive numbers. Determine the following probabilities.

(a) If k balls are randomly withdrawn (without replacement), what is the probability that exactly r of them are white?

(b) If two balls are randomly withdrawn (without replacement), what is the probability that they are the same color?

(c) If k balls are randomly drawn (with replacement, so each time a ball is selected there are n white balls and m black balls) what is the probability that exactly r of them are white?

(d) If two balls are randomly drawn (with replacement, so each time a ball is selected there are n white balls and m black balls) what is the probability that they are the same color?

4. Recall that P(X = k) =p^k(1 - p)^n-k for k = 0, 1, . . . , n is the binomial distribution with parameters n and p. Prove that E[X] = np in any way you like.

5. Suppose we have 10 coins such that if the i^th coin is flipped it will show heads with probability i/10 for i = 1, 2, . . . , 10. A coin is randomly selected and flipped. If it shows heads what is the conditional probability that it is the 7^th coin?

6. Let (X, Y) have joint density f(x, y) = e^-y for 0 < x < y, and f(x, y) = 0 otherwise.

(a) Show that the marginal density of Y is f_Y (y) = ye^-y for y > 0 and 0 otherwise.

(b) Show that fX|Y (x, y) = 1/y for 0 < x < y.

(c) Show that E(X|Y = y) = y/2 for y > 0.

(d) Use part (c) to compute E(X).

7. Let X and Y be independent Poisson random variables with parameters λ₁ and λ₂ respectively. Use moment generating functions to show that the distribution of the sum X + Y is also Poisson and determine the parameter for this sum. The moment generating function for a Poisson random variable with parameter λ is

φ(t) = e^λ(e^t-1) = exp λ(e^t - 1).

8. Suppose that by any time t the number of people that have arrived at a train depot is a Poisson random variable with mean λt. The first train arrives at a time that is uniformly distributed over (0, T) (for some T) independent of when passengers arrive. What is the mean number of passengers that enter the train?

9. The number of people that enter a store in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered the store, given that 10 women entered in that hour. What assumptions have you made?

10. (a) Suppose that the weather today depends on the weather conditions of the last two days. The weather for a given day can be either wet or dry. If it rained the past two days, then it will rain today with probability 0.8. If it did not rain for the past two days then it will rain today with probability 0.3. In any other case, the weather today will, with probability 0.6 be the same as yesterday. Construct an appropriate Markov chain for this model and determine its transition matrix P.

(b) If the weather today depends on the weather of the previous a days and if there are b possible weather conditions each day, how many states are needed to analyze this with a Markov chain?

11. A small barbershop, operated by a single barber, has room for at most two customers. Customers leave if the shop is full when they arrive. Potential customers arrive at a Poisson rate of three per hour, and the successive service times are independent exponential random variables with mean 1/4 hour. What is

(a) the average number of customers in the shop?

(b) the proportion of potential customers that enter the shop?

12. Suppose that a one-celled organism can be in one of two states - either A or B. An individual in state A will change to state B at an exponential rate α; an individual in state B divides into two new individuals of type A at an exponential rate β. Define an appropriate continuous time Markov chain for a population of such organisms and determine all the appropriate parameters for this model.

13: Recall that the long term probabilities of j people being in the system for an M/M/1 queue are π(n) = (λ/µ)ⁿ(1 - λ/µ). Consider an M/M/1 queue in which arrivals finding N people in the system do not enter. What are the long term probabilities for this system?

(II) Proofs-

14: A researcher wants to determine the relative efficacies of two drugs. The results (differentiated between men and women) were as follows:

women	drug I	drug II
success	200	10
failure	1800	190

 

men	drug I	drug II
success	19	1000
failure	1	1000

We are now faced with the question which drug is better. Here are two possible answers.

(1) Drug I was given to 2020 people, of whom 219 were cured. Drug II was given to 2200 people, of whom 1010 were cured. Therefore drug II is much better. (2) Amongst women, the success rate of drug I is 1/10 and for drug II the success rate is 1/20. Amongst men, these rates are 19/20 and 1/2 respectively. In both cases drug I is better. Which of the two answers do you believe? Can you explain the paradox?

15. Answer (c) and only one of (a) or (b):

(a) Show that if X is normally distributed with parameters µ and σ² then Y = aX + b is normally distributed with parameters aµ+b and a²σ². Do this by showing that the cumulative distribution function of Y can be expressed in terms of that of X as F_Y (x) = F_X(x-b/a). (Do not use integrals, just use the definition of the distribution function.) Then differentiate this equation and show clearly how the result is the density of a normal distribution with parameters aµ + b and a²σ².

(b) Show the same result as (a) by first showing that the moment generating functions satisfy M_Y (t) = e^tbM_X(at) and using the fact that M_X(t) = e^µt+σ2t2/2.

(c) If X has normal distribution with mean µ and variance σ² then it has moment generating function M_X(t) = e^µt+σ2t2/2. Use this to show that if X₁ is normal with mean µ₁ and variance σ²₁ and X₂ is normal with mean µ₂ and variance σ²₂ and they are independent then Y = X₁ + X₂ is normal with mean µ₁ + µ₂ and variance σ²₁ + σ²₂.

16. Recall that the geometric distribution with parameter p is given by P(X = n) = p(1 - p)^n-1 for n = 1, 2, . . .. Let X and Y be independent geometric random variables both with parameter p.

(a) Show P(X > t) = (1 - p)^t for t ≥ 1 analytically. That is, show that _n=t+1Σ^∞ p(1 - p)^n-1 = (1 - p)^t.

(b) Show that P(X = m + k|X > m) = P(X = k).

(c) The distribution of X + Y is negative binomial with parameters 2 and p. That is, P(X + Y = k) = (k - 1)p²(1 - p)^k-2 for k = 2, 3, . . .. Show this directly using a formula for the distribution of X + Y in terms of a sum Σ.

(d) Compute E(X|X + Y = k) for all k = 2, 3, . . ..

(e) Explain each of the results above using the interpretation of a geometric random variable as giving the probability of n trials up to and including the first success.

17. The Gamma distribution with parameters α > 0 and λ > 0 has density f(x) = λe^-λx(λx)^α-1/Γ(α) for x ≥ 0 and f(x) = 0 for x < 0 where Γ(α) = ₀∫^∞ e^-yy^α-1 dy.

(a) Show that X has moment generating function M_X(t) = (λ/λ-t)^α.

(b) Show, using any method that you like, that if X₁, X₂, . . . , X_n are independent exponentially distributed random variables with the same parameter λ then X₁ + · · · + X_n has gamma distribution with parameters n and λ.

18. Let T_i, i = 1, 2, . . . , n be independent exponential random variables with parameters λ_i.

(a) Determine (with proof) P(min(T₁, . . . , T_n) > t). Use this to show that the minimum is exponential and determine its rate.

(b) For n = 2, prove that P(T₁ < T₂) = (λ₁/λ₁+λ₂).

(c) Prove that P(T_i = min(T₁, . . . , T_n)) = (λ_i/λ₁+···+λ_n).

19. Consider a Poisson process with rate λ = 10. Given that exactly 5 events occur before time t = 12,

(a) What is the probability that all 5 of the events occur by time s = 9?

(b) What is the probability that exactly 2 of the events occur in the time interval [3, 10]?

(c) In general, for a Poisson process and s < t show that

20. Consider Gambler's ruin. On each turn you win 1 with probability p or lose 1 with probability q = 1 - p. You quit when you reach 0 or N.

(a) Determine with a direct proof (not using martingales) the probability of reaching N before 0 starting at x. That is, the probability of winning. If you can't do this for general p at least do it for p = 1/2.

(b) Writing P(s_i = 1) = p and P(s_i = -1) = 1 - p = q and S_n = S₀ + s₁ + · · · + s_n with g(x) = ((1 - p)/p)^x show that M_n = g(S_n) is a Martingale.

(c) Using the stopping theorem for martingales to generalize the result of part (a), determining the probability of reaching b > a before a.

(III) Overview of the subjects-

21. You find yourself at a graduation party sitting with your father, your grandmother and one of your professors. They ask the following questions. What are your answers?

(a) From your father: We just spent all of this money for you to study probability and stochastic processes. What is it good for?

(b) From your grandmother: You just spent a lot of time studying probability and stochastic processes. What are they?

(c) From your professor: I just spent a lot of time trying to help you understand probability and stochastic processes. What results did you find really interesting and what did you think was a waste of time?