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)=

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

answer:  

step-by-step explanation:

here x represents the height of the pole,

and, the height of the shadow of the pole is 12 feet.

also, the angle of elevation from the top of the pole is 40°,

therefore,

by the low of sine,

which is the required equation for finding the value of x,

the smallest possible measurement of the width is 14 feet

step-by-step explanation:

let's assume length of parallelogram is 'l'

width of parallelogram is 'w'

the perimeter of a parallelogram must be no less than 40 feet

so, we get

we are given

length of parallelogram is 6 feet

so,

l=6

we can plug it

now, we can solve for w

subtract both sides by 12

divide both sides by 2

and we get

so,

the smallest possible measurement of the width is 14 feet