A $1 bet in a state lottery Pick 3 game pays $500 if the 3-digit number you choose exactly matches the winning number, which is drawn at random. Here's the distribution of the payoff X:Payoff X $0 $500Probability 0.999 0.001Each day's drawing is independent of other drawings.a) What's the mean and the standard deviation of X?b)Joe buys a Pick 3 ticket twice a week. What does the law of large numbers say about the average payoff Joe receives from his bets?c) What does the central limit theorem say about the distribution of Joe's average payoff after 104 bets in a year?

Answer:Step-by-step explanation:Given that in a  $1 bet in a state lottery Pick 3 game pays $500 if the 3-digit number you choose exactly matches the winning number, which is drawn at random. Here's the distribution of the payoff X:we find that each game is independent of the other and probability for success in each game p = 0.001 is constantX is binomiala) Mean of X = E(X) = [tex]np = 0.001 n[/tex] where n is no of ticketsVar of (X) = npq = 0.001 npStd dev (X) =[tex]\sqrt{0.001*0.999n} \\=0.0317\sqrt{n}[/tex]b) Here n =3 per weekwhen he purchases more tickets such n tends to infinity sample mean would almost equal to the population mean.i.e. he can expect 0.001 per trial only.c) Central limit theorem says for 104 bets the pay off would be approxy normal with mean = 0.104 and std dev = 0.322