Suppose each day (starting on day 1) you buy one lottery ticket with probability 1/2; otherwise, you buy no tickets. A ticket is a winner with probability p independent of the outcome of all other tickets. Let Ni be the event that on day i you do not buy a ticket. Let Wi be the event that on day i, you buy a winning ticket. Let Li be the event that on day i you buy a losing ticket.
(a) What are P[W33], P[L87], and P[N99]?
(b) Let K be the number of the day on which you buy your first lottery ticket. Find the PMF PK (k).
(c) Find the PMF of R, the number of losing lottery tickets you have purchased in m days.
(d) Let D be the number of the day on which you buy your jth losing ticket. What is PD(d)? you buy your jth losing ticket on day d, how many losers did you have after d - 1 days?