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trasher [3.6K]
3 years ago
8

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, an

d 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)=
Mathematics
1 answer:
omeli [17]3 years ago
5 0

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

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13. The equation of line l is given as y = 0.6x – 4.
Luden [163]

Answer: 16.6

Step-by-step explanation:

a = slope = 0.6

b = y-intercept = -4

c = x intercept = 20/3. (Set y=0 and solve for x)

So, a+b+3c = 0.6 - 4 + 3*20/3 = 16.6

3 0
3 years ago
Find all polar coordinates of point p where p = ordered pair 5 comma pi divided by 3.
Aleonysh [2.5K]

Find all polar coordinates of point p where p = ordered pair 5 comma pi divided by 3.

The polar coordinate of any point can be written as:

<span>(r, θ) = (r, θ + 2nπ)                                           --> when positive</span>

<span> (r, θ) = [ - r, θ + (2n + 1)π ]                           --> when negative</span>

<span> where n is any integer.</span>

The polar coordinates of this given point P is: P = (r, θ) = (5, π/3).

When the value of r is positive, the polar coordinate is written as P= (5, π/3) = (5, π/3 + 2nπ)

where n is any integer.

When the value of r is negative, the polar coordinate is written as P = (5, π/3) = [ - 5, π/3 + (2n + 1)π] where n is any integer.

 

Therefore all polar coordinates of point P are (5, π/3 + 2nπ) and [ - 5, π/3 + (2n + 1)π ].

<span> </span>

7 0
3 years ago
5b - 5 - 12b - 7<br> I need steps please and thank you
Misha Larkins [42]

Answer:

7b-12

Step-by-step explanation:

12b-5b-7-5

7b-12

8 0
3 years ago
In Juan's coin bank there are nickels (n),
alukav5142 [94]

Answer:

The answer is 8 quarters, 16 nickles, and 32 dimes

Step-by-step explanation:

  1. First, the equation would be (2/4d) x 5 +d x 10 +1/4d x 25= 600 (Don't put 6.00, that's dollars. We put 6 dollars into cents to make it easier.)
  2. Then we slowly start to solve: 10/4d + 10d + 25/4d = 600
  3. 10d+40d+25d/4 = 600 (Put all of that over 4, not first 25d)
  4. 75d= 600 x 4
  5. d=600 x 4/75 (same here put everything over 75)
  6. d=32
  7. So then n= 2 x 32/4 = 16
  8. Then n would be equal to : 1/4d= 1/4 x 32 = 8
  9. So your answer would be 8 quarters, 16 nickles, and 32 dimes.

Hope this helped :)

5 0
3 years ago
What is the volume of a hemisphere that
ch4aika [34]

<u>Given</u>:

Given that the diameter of the hemisphere is 12.6 cm

The radius of the hemisphere is given by

r=\frac{d}{2}=\frac{12.6}{2}=6.3

We need to determine the volume of the hemisphere.

<u>Volume of the hemisphere:</u>

Let us determine the volume of the hemisphere.

The volume of the hemisphere can be determined using the formula,

V=\frac{2}{3} \pi r^3

Substituting the values, r = 6.3 and π = 3.14, we have;

V=\frac{2}{3} (3.14)(6.3)^3

Simplifying the values, we have;

V=\frac{2}{3} (3.14)(250.047)

V=\frac{1570.29516}{3}

Dividing, we have;

V=523.43172

Rounding off to the nearest tenth, we get;

V=523.4

Thus, the volume of the hemisphere is 523.5 cm³

6 0
3 years ago
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