Stat 5101 Lecture Notes - School of Statistics

4.4. THE POISSON PROCESS 1254-3. A brand of raisin bran averages 84.2 raisins per box. The boxes are filledfrom large bins of well mixed raisin bran. What is the standard deviation of thenumber of raisins per box.4-4. Let X be the number of winners of a lottery. If we assume that playerspick their lottery numbers at random, then their choices are i. i. d. randomvariables and X is binomially distributed. Since the mean number of winnersis small, the Poisson approximation is very good. Hence we may assume thatX ∼ Poi(µ) where µ is a constant that depends on the rules of the lottery andthe number of tickets sold.Because of our independence assumption, what other players do is independentof what you do. Hence the conditional distribution of the number of otherwinners given that you win is also Poi(µ). If you are lucky enough to win, youmust split the prize with X other winners. You win A/(X + 1) where A is thetotal prize money. Thus( ) AEX +1is your expected winnings given that you win. Calculate this expectation.4-5. Suppose X and Y are independent, but not necessarily identically distributedPoisson random variables, and define N = X + Y .(a)(b)Show thatX | N ∼ Bin(N,p),where p is some function of the parameters of the distributions of X, Y .Specify the function.AssumeZ | N ∼ Bin(N,q),where 0