Every day you consider going jogging. Before each mile, including the first, you will quit with probability q, independent of the number of miles you have already run. However, you are sufficiently decisive that you never run a fraction of a mile. Also, we say you have run a marathon whenever you run at least 26 miles.
(a) Let M equal the number of miles that you run on an arbitrary day. What is P[M > 0]? Find the PMF PM (m).
(b) Let r be the probability that you run a marathon on an arbitrary day. Find r.
(c) Let J be the number of days in one year (not a leap year) in which you run a marathon. Find the PMF PJ (j). This answer may be expressed in terms of r found in part (b).
(d) Define K = M - 26. Let A be the event that you have run a marathon. Find PK|A(k).