Question

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 ? independent of the outcome of all other tickets. Let ?? be the event that on day ? you do not buy a ticket. Let ?? be the event that on day ?, you buy a winning ticket. Let ?? be the event that on day ? you buy a losing ticket.

a. Find the PMF of ?, the number of losing lottery tickets you have purchased in ? days.

b. Let ? be the number of the day on which you buy your ? ?ℎ losing ticket. What is ??(?)? Hint: If you buy your ? ?ℎ losing ticket on day d, how many losing tickets did you have after ? − 1 days?

Answer 1

Part a)The number of ways in which r days(where the losing tickets will occurs) can be chosen among m days is .

Now, if on a day the person has a losing ticket, the probability of this event is P[Li]=(1/2)(1-p).

Therefore, the PMF of R is given by :

P[R=r]=, r=0,1,2,..,m and 0 , otherwise

Part b) Here PD(d) =P[the losing ticket appears on the day]

                                =(1/2)(1-p)P[the number of losing tickets in d-1 days = j-1],

                                =, using the PMF obtained in Part a)