Math 170A - Homework 3 - Due 25 October 2016 1. An insurance company has 1,000 policies. For each policy, there is one claim with probability 0.15% and no claim otherwise independent of other policies. (i) Write out the expression for (but no need to evaluate) the probability that there are at least 2 claims. (ii) Estimate the probability in (i). 2. Let X1 and X2 be independent Poisson random variables with parameters 1 and 2 , respectively. Show that X1 + X2 is a Poisson random variable with parameter 1 + 2 . 3. Let X be the number of times two fair six sided dices need to be rolled (simultaneously) until the sum is 5. Find P(2 X 4). 4. Let 0 < p < 1. Let X be a geometric random variable with parameter p and let Y be a Bernoulli random variable with parameter 1 p. Assume that X and Y are independent. Show that XY + 1 is a geometric random variable with parameter p. 5. Write out the expressions for (but no need to evaluate) the probabilities of the following events. (i) Head appears at least three times when six fair coins are tossed. (ii) A fair six sided dice is rolled 10 times. At least 3 of the results are 1. (iii) A fair six sided dice is rolled 5 times. Exactly 2 of the results are 1. 6. Let 0 < p, q < 1. Let X be a binomial random variable with parameters (10, p) and let Y be a Bernoulli random variable with parameter q. Assume that X and Y are independent. Find P(X + Y = 5). 1