Exercise 4.28

Answers

Let Ik be the indicator variable for the k-th location, so that Ik = 1 if k-th location has a treasure and Ik = 0 otherwise.

Let X be number of locations William checks to get t treasures, and Xj ∼HGeom(t − 1,n − 1 − (t − 1),j) be the number of treasures found within j checked locations.

By symmetry. P(Ik = 1) = t n. Then,

P(X = k) = P(Ik = 1)P(Xk−1 = t − 1) = t n (t−1 t−1)( n−1−t+1 k−1−t+1) (n−1 k−1) = t n (n−t k−t) (n−1 k−1)

E(X) = ∑ k=tnkP(I k = 1)P(Xk−1 = t − 1) = ∑ k=tn(k t n (n−t k−t) (n−1 k−1) ) = (n + 1)t t + 1

User profile picture
2021-12-05 00:00
Comments