Pattern 2: How many squares?

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

omeli [17]

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

5 0

2 years ago

The function C(x)=−20x+1681 represents the cost to produce x items. What is the least number of items that can be produced so th

Reptile [31]

Answer:

<em>The least number of items to produce is 41</em>

Step-by-step explanation:

<u>Average Cost</u>

Given C(x) as the cost function to produce x items. The average cost is:

$\displaystyle \bar C(X)=\frac{C(x)}{x}$

The cost function is:

$C(x) = -20x+1681$

And the average cost function is:

$\displaystyle \bar C(X)=\frac{-20x+1681}{x}$

We are required to find the least number of items that can be produced so the average cost is less or equal to $21.

We set the inequality:

$\displaystyle \frac{-20x+1681}{x}\le 21$

Multiplying by x:

$-20x+1681 \le 21x$

Note we multiplied by x and did not flip the inequality sign because its value cannot be negative.

Adding 20x:

$1681 \le 21x+20x$

$1681 \le 41x$

Swapping sides and changing the sign:

$41x \ge 1681$

Dividing by 41:

$x\ge 41$

The least number of items to produce is 41

8 0

2 years ago

The mean temperature for the first 4 days in January was 1°C.

soldier1979 [14.2K]

-4 degrees I think.

5 0

2 years ago

HURRY INVERSE OF A FUCTION

Anon25 [30]

Here is how we get the answer....
First, replace f(x) with y . ...
Replace every x with a y and replace every y with an x .
Solve the equation from Step 2 for y . ...
Replace y with f−1(x) f − 1 ( x ) . ...
Verify your work by checking that (f∘f−1)(x)=x ( f ∘ f − 1 ) ( x ) = x and (f−1∘f)(x)=x ( f − 1 ∘ f ) ( x ) = x are both true.

6 0

2 years ago

What is 18 divided by 0.2 and how?

kati45 [8]

0.2 is the same as 2/10

18/(2/10)

As you may have learned in school, when you divide a fraction, you flip the fraction in the denominator, and change it to multiplication

18/(2/10) becomes 18 x (10/2)

multiply normally

18 x 10 = 180

divide to find answer

180/2 = 90

90 is your answer

hope this helps

Remember, when you meet this problem, flip the second fraction, and change the division sign into multiplication. (always second, never first (if first is a fraction)).

5 0

2 years ago