Lent 2017 JRN MATHEMATICAL TRIPOS: PART IA PROBABILITY Example Sheet 2 (of 4) 1. A coin with probability p € [0, 1] of heads is tossed n times. Let E be the event 'a head is obtained on the first toss' and Fk the event 'exactly k heads are obtained'. For which pairs of non-negative integers (n, k) are E and Fk independent? 2. The events A and B are independent. Show that the events Ac and B are independent, and that the events Ac and Be are independent. 3. Independent trials are performed, each with probability p of success. Let Pn be the probability that n trials result in an even number of successes. Show that Pn = +(1 + (1 - 2p)"). 4. Two darts players A and B throw alternately at a board and the first to score a bull wins the contest. The outcomes of different throws are independent and on each of their throws A has probability pA and B has probability pp of scoring a bull. If A has first throw, calculate the probability p that A wins the contest. 5. Consider the probability space 2 = {0, 1}3 with equally likely outcomes. (a) Show that there are 70 different Bernoulli random variables of parameter 1/2 that can be defined on 2. (b) How many Bernoulli random variables of parameter 1/3 can be defined on ( ?? (c) What is the length of the longest sequence of independent Bernoulli random variables of pa- rameter 1/2 that can be defined on 12? 6. Suppose that X and Y are independent Poisson random variables with parameters À and u re- spectively. Find the distribution of X + Y. Prove that the conditional distribution of X, given that X +Y = n, is binomial with parameters n and >/(A + p). 7. The number of misprints on a page has a Poisson distribution with parameter A, and the numbers on different pages are independent. (a) What is the probability that the second misprint will occur on page r? (b) A proof-reader studies a single page looking for misprints. She catches each misprint (indepen- dently of others) with probability p E [0, 1]. Let X be the number of misprints she catches and let Y be the number she misses. Find the distributions of the random variables X and Y and show they are independent. 1
n X= 1 n i=1 >Xi and S2 = >(Xi-X)2. i=1 n 8. Let X1, ... , Xn be independent identically distributed random variables with mean u and variance 02. Find the means of the random variables 9. In a sequence of n independent trials the probability of a success at the ith trial is pj. Let N denote the total number of successes. Find the mean and variance of N. 10. Liam's bowl of spaghetti contains n strands. He selects two ends at random and joins them together. He repeats this until no ends are left. What is the expected number