Every day you consider going jogging. Before each mile, including the first, you

Question

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. Let M equal the number of miles that you run on an arbitrary day. What is P[M > 01? Find the PMF P_M (m). Let r be the probability that you run a marathon on an arbitrary day. Find r. Let J be the number of days in one year (not a leap year) in which you run a marathon. Find the PMF P_j(j). This answer may be expressed in terms of r found in part (b). Define K = M - 26. Let A be the event that you have run a marathon. Find Pk|A(K).

Explanation / Answer

(a) Indipendent events every day P( quit before one mile) = P(0) = q

so P [ M>0 ] = 1 - q

PMF Pm(m) = P ( running m miles) * P( quitting before m+1 miles) = q(1- q)m will be calculated when he will run m miles and stop before ( m+1) th mile.

(b) i will run a maratheon on an arbitary day when P ( m>26) = q ( 1-q)26 + q ( 1-q)27 + ...........

r = (1-q)26

(c) J be the number of days in one year = 365 days in which i will run a maratheon .

so PMF PJ (j) = 365CJ (r) J( 1-r) 365 -J

(d) K = M - 26

P(K/A)(k) = Here ik means that how much more miles he run above 26 KM

so P(K/A)(k)  = q(1- q)m/r

by putting value of m = K + 26 and r = (1-q)26

P(K/A)(k) = q(1- q)m/r = q (1-q)K +26/ (1-q)26= q(1 -q)K

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