Answer:
1. p*(1-p)
2. n*p*(1-p)
3. p*(1-p)
4. 0
5. p^2*(1-p)^2
6. 57/64
Step-by-step 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
