Question

Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let R be the sum of the rewards collected at times 1, 2,...,? ER and Var (R) by carrying We will find notation (available through the "STANDARD NOTATION button below.) Remember to write for all multiplications and parentheses vwhere necessary. out a sequence of steps. Express your answers below in terms of p andfor n using standard to include We first work towards finding ER. 1. Let I denote the reward tpossibly 0) given at time k, for k E 1,2, ... , n} Find = 2. Using the answer to part 1, find E R E며 The variance calculation is more involved because the random variables , J..., I following values. are not independent. We begin by computing the 3. If e 1,2,..,n} then E 4. If k E 1,2,..,n -1}then 5. If &1, 2 and k n then E[ 3/4 n 10 6. Using the results above, calculate the numerical value of Var (R) assuming that p = Var (R

Answer 1

Let Xk denote the indicator variable that at kth toss, head has appeared The expectation of reward at kth toss is (1) E 1P(X 0, X-1= 1)0 — р(1 — р) From part a) (2) we get that E[R E пр(1 — р) Expectation of is (3) E 1P(Xk 0, Xk-1 = 1)0 — p(1 — р) (4) Observe that if / = 1, then at kth toss, tail has appeared. Hence Ik+1 Similarly, k 1 => I = 0. Hence, EIIk+1] = 0 0 independent for >2 (5) Since the random variables X4 and Xk+t-1 are hence E[Xk]E[Xk+] = p2(1 - p)2 E[XXk+ From part (5) and (6) (6) get we JpP(1- р)? if1 -1 Cov(Xk, Xk+t if 2, k <n Var(Xk) EX- (EX) — p(1 — р) — р? (1 -р)? — Р(1— Р)1 — р(1 — р)] Hence k (k) Var(R) Var i-1 k kn-k Var(Ik) Cov(Xh,Xkt) k =p1-p) -p(1 - p)] - p(1-p)2 i=1 - пр(1 — р)1 — 2р(1 — р)]