Suppose that a sequence of independent tosses are made with

Suppose that a sequence of independent tosses are made with a coin for which the probability of obtaining a head on each given toss is 1=30. (a) What is the expected number of tails that will be obtained before ve heads have been obtained? (b) What is the variance of the number of tails that will be obtained before ve tails have been obtained? (c) What is the expected number of tosses that will be required in order to obtain ve heads? (d) What is the variance of the number of tosses that will be required in order to obtain ve heads?

Solution

Let X denote number of tails required to get 5 heads.

Then X~NB(5,p) where p is the probability of getting head.

We know E(X) = 5(1-p)/p   and V(X) = 5(1-p)/p²

Given that p=1/30

(a)

E(X) = 5*(1-1/30) / (1/30) = 145

(b)

V(X) = 4350

(c)

Let Y be the number of tosses required to get 5 heads.

Then Y=X+5

E(Y) = E(X+5) =E(X) + 5 = 145 +5 =150

(d)

V(Y) = V(X+5) =V(X) = 4350   (Since variance is independent of change of origin).