Question

Problem 9.1 A gatekeeper stands in front of a locked door. He has a keychain with 12 keys and a single one of them unlocks the door in front of him. (a) First assume that he tries the keys one after the other. Let X be the random variable that describes the number of unsuccessful attempts at unlocking the door in this case. (i) Let X(12) denote all the possible values that X can take. Determine X(22). (ii) What is the probability that P(X = 0)? P(X = 1)? P(X = k) for any k e X(22)? (iii) Determine the expected number of failed attempts. (b) Assume now that it is very cold and dark. The gatekeeper therefore drops the keychain after each failed attempt, and he no longer knows which key he has already tried. Let Y be the random variable that describes the number of unsuccessful attempts at unlocking the door in that case. (i) Determine the distribution of Y. (ii) What is the cumulative distribution function of Y? (iii) What is the probability that he will need more than 12 attempts? (iv) What is the expected number of attempts? (v) What is the expected number of failed attempts?

Answer 1

a Xal = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 112 i P(x=0 = 1 12 P(x=1) = 11 x 1 = 21 11 12 12 27 144 P(x = k) = (1.11 k-1 - 11k 12 12k+1 P(x=) = 42 12 ui, Expected value. E(X) = Ex P(X=2) 2 - = E x 112 12 12 1 12012 12) x=0

= E(X) = 11 11 x 1 x 34.85 12 12 > E(X) = 2.622 be y follows geometric distribution with parametel p=1 12 So Pyly) = 1 x (11) 12 where k=0, 1, 2,- to solve 4 Note: We are legally bounded parts of the problem. Thank You