Question

Three friends, Xena, Yvonne, and Zelda, decide to run the Boston marathon. For each of them the time required to complete the marathon is a continuous random variable uniformly distributed between 4 hours and 6 hours. The running times of all contestants are independent.

After the marathon, a one­ hour long TV show interviews the three friends, with 1/2 of the time devoted to the winner, and 1/3 and 1/6 to the second and third, respectively. You are at home, and you do not know the results of the marathon. You want to find out who won, so you turn on the TV at a random time during the show. You see Zelda on the TV screen.

(a) What’s the probability that Zelda won the race? What’s the probability density function of Zelda’s race time?

(b) Does the probability of Zelda being the winner change, if you observe her talking non-­stop for five minutes? How does the probability distribution of her running time change?

Note: If two or more contestants arrive at exactly the same time, the ranking is decided by random tie­breaking, with equal probabilities.

Answer 1

Gluen daa uoul Xena x Y- Yvonne · 2- Xelda G-4 able 3 eien hauue

6 and elda - 3 elda onTV on-tv etda on a*c