Question

consider a sequence of 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.A reward of one unit is given at time k, for k∈{1,2,…,n}, if the toss at time k resulted in Tails and the toss at time k−1 resulted in Heads. Otherwise, no reward is given at time k.Let R be the sum of the rewards collected at times 1,2,…,n.We will find E[R] and var(R) by carrying out a sequence of steps. Express your answers below in terms of p and/or n using standard notation. Remember to write '*' for all multiplications and to include parentheses where necessary.We first work towards finding E[R].1. Let Ik denote the reward (possibly 0) given at time k, for k∈{1,2,…,n}. Find E[Ik].E[Ik]=2. Using the answer to part 1, find E[R].E[R]=The variance calculation is more involved because the random variables I1,I2,…,In are not independent. We begin by computing the following values.3. If k∈{1,2,…,n}, thenE[I2k]=4. If k∈{1,2,…,n−1}, thenE[IkIk+1]=5. If k≥1, ℓ≥2, and k+ℓ≤n, thenE[IkIk+ℓ]=6. Using the results above, calculate the numerical value of var(R) assuming that p=3/4, n=10.var(R)=

Answer 1

1. p*(1-p)

2. n*p*(1-p)

3. p*(1-p)

4. 0

5. p^2*(1-p)^2

6. 57/64

Explanation:

1. Let Ik denote the reward (possibly 0) given at time k, for k∈{1,2,…,n}. Find E[Ik].

E[Ik]= p*(1-p)

2. Using the answer to part 1, find E[R].

E[R]= n*p*(1-p)

The variance calculation is more involved because the random variables I1,I2,…,In are not independent. We begin by computing the following values.

3. If k∈{1,2,…,n}, then

E[I2k]= p*(1-p)

4. If k∈{1,2,…,n−1}, then

E[IkIk+1]= 0

5. If k≥1, ℓ≥2, and k+ℓ≤n, then

E[IkIk+ℓ]= p^2*(1-p)^2

6. Using the results above, calculate the numerical value of var(R) assuming that p=3/4, n=10.

var(R)= 57/64