Question

Each week, a graduate student named Eliza gives a copy of her job market paper to each of three professors. Each professor then randomly assigns a score to the paper, where the score is an integer ranging from one to four. If the sum of the three scores is 11 or higher, Eliza is allowed to present her paper at that week is seminar. If the combined score is less than 11, she must wait one week and try again. Scores assigned by different professors are independent, and scores assigned by a given professor are independent across weeks. Denote X as the number of weeks until Eliza presents her paper (we also count the week of her presentation). Derive the pdf of X.

Answer 1

as scores from a professor are uniformly distributed frm 1to 4 ; therefore P(getting a score from a professor )=1/4

here probability that Eliza gets a score of 11 or more =P(Eliza gets a score of 11)+P(Eliza gets a score of 12)

=P(2 of professor gives a score of 4 and one score gives a score of 3)+P(all three professor gives a score of 4)

=³C₂(1/4)³+(1/4)³ =(3/64)+(1/64)=4/64 =1/16

hence X follows geometric distribution with parameter p=1/16

pd fof X :f(x)=P(X=x) =P(getting a score less then 11 on x-1 weeks and score of 11 or higher on xth week)

f(x) =(15/16)^x-1*(1/16)