October 1st, 2007

Introduction to Probability

Nathanael Berestycki

Example Sheet 1, Michaelmas Term 2007.

1. Let be a set and let F be a sigma-field on . For j 1, let Aj F . Show thatj1Aj F .

2. Let be a set and let (Bi)iI be a collection of events of , where I is someindex set. Show that the intersection F of all sigma-fields that contain (Bi)iI isa sigma-field. It is called the sigma-field generated by (Bi)iI and is often denotedby (Bi)iI . It is the smallest sigma-field containing all the events (Bi)iI .

3. Let be a set and F be a sigma-field. Let A be an event. What is the sigma-fieldgenerated by A?

4. Let (,F , P ) be a probability space. (a) Show that the probability that exactlyone of the events E or F occurs (not both) is equal to P (E) +P (F ) 2P (E F ).(b) Show that P (E F ) = 1 P (E) P (F ) + P (E F ).

5. (a) You are given a conventional deck of cards. What is the probability that thetop card is an ace?

(b) You count a deck of cards (face down) and find it defective, having only 49cards. What is the probability that the top card is an ace?

6. In this problem, explain all factors arising in your solution. Compute the proba-bility that a poker hand contains:

(a) one pair (aabcd with a, b, c, d distinct card ranks (face values); answer: 0.4226)

(b) two pairs (aabbc with a, b, c distinct card ranks (face values); answer: 0.04754)

7. Let A be an event such that A and A are independent. Show that P (A) = 0 or1. More generally, let A and B be mutually exclusive. Are A and B independent?Explain your answer fully.

8. Bayes rule. Let (,F , P ) be a probability space. Let A be an event, and let(Bi, 1 i n) be a partition of with Bi F , 1 i n. Prove that

P (B1|A) = P (B1)ni=1 P (Bi)P (A|Bi)

P (A|B1)

9. 5% of men and 0.25% of women are colourblind. What is the probability that acolourblind person is a man ? One may use Problem 8 in the solution.

10. Binary digits (0s and 1s) are sent through a noisy communication channel. Asthey are transmitted a mistake can occur with probability > 0 (a 0 will bechanged to a 1 and conversely).

(a) If 0s and 1s are equally likely, what is the probability that the digit sent wasa 1 given that we received a 1 ?

(b) To improve the reliability of the channel, we repeat each digit three times.What is the probability that the message sent was 111 given that we received101 ? Given that we received 000 ? One may use Problem 8 in the solution.

11. Let S be a fixed set of cardinality n. Let A and B be two independent randomsubsets of S, chosen uniformly among all 2n possible subsets of S.

(a) Show that P (A B) = (3/4)n. (Hint: condition on the cardinality of B.)(b) Show that P (A B = ) = (3/4)n.

12. You have a coin that you suspect is not fair. To test it, you toss the coin n timesand you observe k Heads and (n k) Tails, where 1 k n.(a) Let 0 < p < 1. Compute the probability, for a coin that comes up Heads with

probability p, that in n independent tosses you will observe k Heads.

(b) Let f(p) denote the probability obtained in (a). Determine the maximumlikelihood estimator for p, i.e., the value of p which maximizes the probabilityf(p).

13. Let X, Y be independent Poisson random variables, with respective parameters and . Show that X + Y is Poisson with parameter + . Use this to find theconditional distribution of X given that X + Y = n, for some n 1.

14. Let X be an integer-valued random variable, with X 0 almost surely. Show thatE(X) =

n=0 P (X > n).

15. A man is trying to sell his car. He receives successive offers denoted by X0, X1, . . ..After seeing the first offer X0, our man decides to take the first offer that is strictlygreater than X0. Let T = inf{n 1 : Xn > X0}. Assuming that (Xi, i 0)are i.i.d. random variables with a given absolutely continuous distribution, arguethat E(T ) = . (Hint: compute P (T > n) and use the previous problem).Consequence: if you are in a hurry, you are better off taking the first offer!

16. Coupon collector problem. There are n 1 tickets in a box labelled 1, . . . , n.We sample from the box with replacement until we have collected all tickets. Thatis, each time we draw a ticket at random, note the number on the ticket, and putit back in the box. Let N be the number of draws before we have collected alln tickets. (Formally, (Xi, i 1) are i.i.d uniform random variables on {1, . . . , n},and N = inf{k 1 : #{X1, . . . , Xk} = n}). Show that

E(N) = nni=1

1

i

and that

Var(N) = nn1j=0

j

(n j)2 .

(Hint: Consider Xi the number of draws necessary to collect the ith ticket. What

type of random variable is Xi ?)

17. Let X be a real random variable and let F be its cumulative distribution function.Show that (a) F is nondecreasing, (b) limt F (t) = 0 and (c) limt F (t) = 1.Finally, show that (d) F is right-continuous and has left limits. Conclude that Fis continuous at t if and only if P (X = t) = 0.

18. Conversely, show that a real-valued function which satisfies (a),...,(d), is the cdfof a real random variable. (Hint: consider G(t) = inf{s R : F (s) > t}, and letX = G(U), where U is a uniform random variable on (0, 1).)

19. (a) Let X be an exponential random variable with rate > 0. Show that X ismemoryless : that is, for any s, t 0, P (X > t+ s|X > t) = P (X > s).

(b) A line of 10 people is waiting at a post office, where there are three clerks. Eachtime a customer is served, the next customer in line goes to the available clerk.You are last in line. Assuming that each customer requires an independentexponential amount of time (with parameter > 0 identical for all threecashiers), let p be the probability that you are the last one, among the 50persons initially in line, to leave the post office. Show that p is only 1/3.