Reduce the following 4 x 4 game matrix to find the optimal strategy for the row player 4 3 9 7 -7 -5 -3 5 -1 4 5 8 3-5 -1 5 1 (57601/60) 10 5/6 1/60) always play row 2 always play row 3

The proof of expression is shown below.

We have to given that;

sec theta + tan theta = m

To prove,

⇒ cosec θ = (m² - 1) / (m² + 1)  .. (ii)

Now, From expression ,

sec θ + tan θ = m

1/cos θ + sin θ /cos θ = m

(1 + sin θ) / cos θ = m

Plug the value of θ in (ii);

⇒ cosec θ = ((1 + sin θ) / cos θ )² - 1) / ((1 + sin θ) / cos θ )² + 1)

⇒ cosec θ = (1 + sin θ)² - cos²θ / (1 + sin θ)² + cosθ²

⇒ cosec θ = cosecθ

Thus,  The proof of expression is shown

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Answer: 6π cm

Step-by-step explanation:

The formula for the area of a circle is:

A = πr²

where A is the area and r is the radius.

Given that the area of the circle is 9π cm², we can solve for the radius as follows:

9π = πr²

Dividing both sides by π, we get:

r² = 9

Taking the square root of both sides, we get:

r = 3

Therefore, the radius of the circle is 3 cm.

The formula for the circumference of a circle is:

C = 2πr

Substituting the value of r, we get:

C = 2π(3) = 6π

Therefore, the circumference of the circle is 6π cm.

(i) The pdf of random variable X is:

[tex]fx(x) = {2bx e^{(-bx^2)}[/tex], x > 0

0, x < 0

(ii) The pdf of Y is:

[tex]fy(y) = b\sqrt y / e^{(by)} , y > 0[/tex]

0, y ≤ 0

(i) To find the probability density function (pdf) of X, we need to take the derivative of the cumulative distribution function (cdf) with respect to x.

For x > 0, we have:

[tex]Fx(x) = 1 - e^{(-bx^2)}[/tex]

Differentiating both sides with respect to x gives:

fx(x) = d/dx Fx(x) = [tex]d/dx [1 - e^{(-bx^2)}] = 2bx e^{(-bx^2)}[/tex]

For x < 0, we have:

Fx(x) = 0

Differentiating both sides with respect to x gives:

fx(x) = d/dx Fx(x) = d/dx [0] = 0

Therefore, the pdf of X is:

[tex]fx(x) = {2bx e^{(-bx^2)}[/tex], x > 0

{0, x < 0

(ii) To find the pdf of [tex]Y = X^2[/tex], we can use the transformation method. The transformation function is [tex]g(x) = x^2[/tex].

We have:

Fy(y) = P(Y ≤ y) = P([tex]X^2[/tex] ≤ y) = P(-√y ≤ X ≤ √y) = Fx(√y) - Fx(-√y)

Differentiating both sides with respect to y gives:

fy(y) = d/dy Fy(y) = d/dy [Fx(√y) - Fx(-√y)]

= (1/2y) fx(√y) - (-1/2y) fx(-√y)

[tex]= (1/2y) 2b\sqrt y e^{(-by)}[/tex]

= [tex]b\sqrt y / e^{(by)}[/tex]

Therefore, the pdf of Y is:

[tex]fy(y) = b\sqrt y / e^{(by)} , y > 0[/tex]

0, y ≤ 0

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In 5 possible schedules, John can play basketball on Wednesday or Friday or both.

There are two possible scenarios:

1) John plays basketball on Wednesday only or Friday only, but not both.
- If John plays basketball on Wednesday, he has 2 options left to play on Friday (either play or not play), so there are 2 possibilities.
- If John plays basketball on Friday, he has 2 options left to play on Wednesday, so there are 2 possibilities.
Therefore, there are a total of 2+2=4 possibilities for playing basketball on either Wednesday or Friday, but not both.

2) John plays basketball on both Wednesday and Friday.

In this case, there are only 1 possibility.
So, the total number of possible schedules to play basketball on Wednesday or Friday or both is 4+1=5.

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In 5 possible schedules, John can play basketball on Wednesday or Friday or both.

There are two possible scenarios:

1) John plays basketball on Wednesday only or Friday only, but not both.
- If John plays basketball on Wednesday, he has 2 options left to play on Friday (either play or not play), so there are 2 possibilities.
- If John plays basketball on Friday, he has 2 options left to play on Wednesday, so there are 2 possibilities.
Therefore, there are a total of 2+2=4 possibilities for playing basketball on either Wednesday or Friday, but not both.

2) John plays basketball on both Wednesday and Friday.

In this case, there are only 1 possibility.
So, the total number of possible schedules to play basketball on Wednesday or Friday or both is 4+1=5.

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a) tn-1,alpha/2 = -1.796 (for the lower bound) and tn-1,1-alpha/2 = 1.796 (for the upper bound).

b) tn-1,alpha/2 = -2.447 (for the lower bound) and tn-1,1-alpha/2 = 2.447 (for the upper bound).

c) tn-1,alpha/2 = -12.706 (for the lower bound) and tn-1,1-alpha/2 = 12.706 (for the upper bound).

d) tn-1,alpha/2 = -2.048 (for the lower bound) and tn-1,1-alpha/2 = 2.048 (for the upper bound).

To find the value of tn-1,alpha/2, we need to use a t-distribution table or a statistical software that can calculate critical values.

a) For a 90% confidence interval with sample size n=12, we have n-1 = 11 degrees of freedom. Using a t-distribution table or a statistical software, we find that the critical value for alpha/2 = 0.05 is 1.796. Therefore, tn-1,alpha/2 = t11,0.05/2 = -1.796 (for the lower bound) and t11,1-0.05/2 = 1.796 (for the upper bound).

b) For a 95% confidence interval with sample size n=7, we have n-1 = 6 degrees of freedom. Using a t-distribution table or a statistical software, we find that the critical value for alpha/2 = 0.025 is 2.447. Therefore, tn-1,alpha/2 = t6,0.025/2 = -2.447 (for the lower bound) and t6,1-0.025/2 = 2.447 (for the upper bound).

c) For a 99% confidence interval with sample size n=2, we have n-1 = 1 degree of freedom. Using a t-distribution table or a statistical software, we find that the critical value for alpha/2 = 0.005 is 12.706. Therefore, tn-1,alpha/2 = t1,0.005/2 = -12.706 (for the lower bound) and t1,1-0.005/2 = 12.706 (for the upper bound).

d) For a 95% confidence interval with sample size n=29, we have n-1 = 28 degrees of freedom. Using a t-distribution table or a statistical software, we find that the critical value for alpha/2 = 0.025 is 2.048. Therefore, tn-1,alpha/2 = t28,0.025/2 = -2.048 (for the lower bound) and t28,1-0.025/2 = 2.048 (for the upper bound).

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The given question is incomplete, the complete question is:

Find the value of tn-1,alpha needed to construct anupper or lower confidence bound in each of the situationsin excercise 1.

Excercise 1 says" Find the value of tn-1,alpha/2 needed toconstruct a two-sided confidence interval of the given level withthe given sample size:

a)Level 90% sample size 12.

b)Level 95% sample size 7.

c)Level 99% sample size 2.

d)Level 95% sample size 29.

In this case, the highest degree term is n^7 in the numerator and n^3 in the denominator. Therefore, as n approaches infinity, the sequence grows without bound and diverges. So the answer is "diverges".

To determine if the sequence converges or diverges and find the limit, we'll analyze the given sequence a_n = n^4 / (n^3 - 9n).
Step 1: Identify the highest power of n in both the numerator and the denominator. In this case, it's n^4 in the numerator and n^3 in the denominator.
Step 2: Divide both the numerator and the denominator by the highest power of n found in the denominator, which is n^3.
a_n = (n^4 / n^3) / ((n^3 - 9n) / n^3)
Step 3: Simplify the expression.
a_n = (n) / (1 - (9/n^2))
Step 4: Take the limit as n approaches infinity.
lim n→∞ a_n = lim n→∞ (n) / (1 - (9/n^2))
As n approaches infinity, the term (9/n^2) approaches 0 since the denominator grows much faster than the numerator.
lim n→∞ a_n = lim n→∞ (n) / (1 - 0)
Step 5: Evaluate the limit.
lim n→∞ a_n = ∞

Since the limit goes to infinity, the sequence diverges. Therefore, the answer is "diverges." To determine whether the sequence converges or diverges, we can look at the highest degree term in the numerator and denominator.

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The numeric difference between a sample statistic and a population parameter is called: sampling error. A sample statistic is an estimate based on a portion of the population, while the population parameter is the true value for the entire population. The difference between these two values, known as the sampling error, occurs due to the variation in samples taken from the population.

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The total area of the flower gardens is x(31 - 2x)/2 sq.ft.

(a) The area of the driving region is the area of a rectangle with length 50 ft and width 31 - 2x ft, where x is the length of one side of a parking space.

Since the parking spaces are congruent parallelograms, they can be divided into two congruent right triangles.

The base of each right triangle is x ft, the height is half of the width of the driving region, which is (31 - 2x)/2 ft.

The area of each parking space is the sum of the areas of the two congruent right triangles.

Therefore,

The total area of the surface to be paved is:

Area = Area of driving region + 4(Area of parking space)

= (50 ft) x (31 - 2x ft) + 4[2(x/2 ft) x ((31 - 2x)/2 ft)]

= 1550 - 100x + 2x(31 - 2x)

= 4[tex]x^2[/tex] - 100x + 1550 sq.ft.

(b) The unpaved areas for flowers are congruent triangles each with base x ft and height (31 - 2x)/2 ft.

Therefore,

The total area of the flower gardens is:

Area = 2(Area of one triangle)

= 2[(x ft) x ((31 - 2x)/2 ft)/2]

= x(31 - 2x)/2 sq.ft.

The factor of 2 in the formula.

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So when x = 6 , y = 4.

Given

x=3 when y=8

To Find

The function that models the inverse variation.

Solution

if  x and y vary inversely, we can use the formula:

xy = k

where k is a constant. We can solve for k using the initial condition x = 3 when y = 8:

3(8) = k

k = 24

So the equation that models the inverse variation is:

xy = 24

We can use this equation to find the value of y for a given value of x, or the value of x for a given value of y. For example, if we want to find y when x = 6:

(6)y = 24

y = 4

So when x = 6, y = 4

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Answer:

To write a rational expression with denominator 6b that is equivalent to a/b, we can multiply both the numerator and denominator of a/b by 6 to get:

(a/b) x (6/6) = (6a)/(6b)

Now we have a rational expression with denominator 6b that is equivalent to a/b.

Step-by-step explanation:

Answer: Zahid obtained 170 marks.

Step-by-step explanation:

Let's start with the basic rules of the question.

We know that for each question answered correctly, 5 marks will be given. And for each incorrect answer, 3 marks will be deducted. Now the problem says that Zahid's marks have been deducted by 18. There are 3 marks deducted for each wrong answer so we'll divide 18 by 3, which gives us 6. Zahid got 6 questions wrong. However, there are 40 questions in the exam, so if we assume that the only ones he answered incorrectly are the 6 questions, then we should subtract 6 from 40. This leaves us with only the correct answers left which is 34. Now again, we know that for each correct answer 5 marks will be given. Assuming that Zahid answered the rest of the questions correctly, we should multiply 34 by 5, which gives us 170.

In numbers your workings might look like this:

18 ÷ 3 = 6

40 - 6 = 34

34 × 5 = 170

I hope this helped you answer your problem. Please let me know if you need any further explanation :)

Answer:

it can be written as 0.44y=40

44% can be written as 0.44

Step-by-step explanation:

to solve for y divide both sides by 0.44

to get y is equal to 100

The Z score for a sample mean being 180  is -2.50.

To calculate the z-score for a sample mean x = 180 with a population mean (μ) of 200 and a standard deviation (σ) of 24, we need to use the following formula:

z = (x - μ) / (σ / √n)

In this case, x = 180, μ = 200, σ = 24, and n = 9.

Step 1: Subtract the population mean from the sample mean: (180 - 200) = -20.
Step 2: Divide the standard deviation by the square root of the sample size: (24 / √9) = 24 / 3 = 8.
Step 3: Divide the result from Step 1 by the result from Step 2: (-20) / 8 = -2.5.

The z-score for a sample mean x = 180 is -2.50.

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To find the critical numbers of the function g(x), we need to first find its derivative and then set the derivative equal to zero to solve for x.

The function is given as: g(x) = (64 - x^2)^(1/3)

To find the derivative, we will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

So: g'(x) = (1/3)(64 - x^2)^(-2/3) * (-2x)

Now, we need to set g'(x) = 0 to find the critical numbers:

0 = (1/3)(64 - x^2)^(-2/3) * (-2x)

To solve for x, we can observe that if either of the factors is equal to 0, then the equation will hold.

So, let's examine each factor: (1/3)(64 - x^2)^(-2/3) = 0:

This factor can never be zero, because a nonzero number raised to any power is never zero. -2x = 0: This factor is zero when x = 0.

So, the only critical number for the function g(x) is x = 0. The final answer is: 0

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To find the critical numbers of the function g(x), we need to first find its derivative and then set the derivative equal to zero to solve for x.

The function is given as: g(x) = (64 - x^2)^(1/3)

To find the derivative, we will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

So: g'(x) = (1/3)(64 - x^2)^(-2/3) * (-2x)

Now, we need to set g'(x) = 0 to find the critical numbers:

0 = (1/3)(64 - x^2)^(-2/3) * (-2x)

To solve for x, we can observe that if either of the factors is equal to 0, then the equation will hold.

So, let's examine each factor: (1/3)(64 - x^2)^(-2/3) = 0:

This factor can never be zero, because a nonzero number raised to any power is never zero. -2x = 0: This factor is zero when x = 0.

So, the only critical number for the function g(x) is x = 0. The final answer is: 0

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The joint density function for two variables x and y is denoted by f(x,y). In this case, the joint density function for x and y is given by f(x,y)={23(2)0 for 0≤x≤1,0≤y≤1, otherwise.

This means that the probability of both x and y falling within the given range is proportional to 23(2)0. The density function for a single variable, say x, is obtained by integrating f(x,y) over y. Similarly, the density function for y can be obtained by integrating f(x,y) over x. The expected value of a function of x and y, say g(x,y), denoted by E[g(x,y)], is given by the double integral of g(x,y) times f(x,y) over the region of x and y where f(x,y) is non-zero.

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The correct answer for the above question will be, Option D) Confidence interval will become wider if the sample size is decreased.

The standard error of the mean grows as the sample size decreases. The standard error of the mean is a measure of the variability of sample means that is proportional to sample size. The standard error increases as the sample size decreases, resulting in a broader confidence interval. As a result, when the sample size decreases, the confidence interval grows broader.

A confidence interval is a set of values that, with a high degree of certainty, include the real population parameter. It is determined by taking into account the sample size, standard deviation, and degree of confidence. The broader the confidence interval, the less exact the population parameter estimate.

Therefore, Option D. Confidence interval will become wider if the sample size is decreased is the correct answer.

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Answer:

id go 48 The circumference is 16π cm, about 50.27 cm.

Step-by-step explanation:

diameter: 16 cm

circumference: 16π cm ≈ 50.27 cm

Step-by-step explanation:

The diameter is twice the radius:

 d = 2r = 2(8 cm)

 d = 16 cm

The diameter is 16 cm.

__

The circumference is pi times the diameter.

 C = πd

 C = π(16 cm)

 C = 16π cm ≈ 50.27 cm

The answers for questions a,b, and c involving an exponential random variable, probability, and conditional probability are P(X > 5) = e^(-a * 5), P(X > 5 | X > 2) = (e^(-a * 5)) / (e^(-a * 2)), and P(4 <= X <= 4 + 2δ | X > 2) = f(4) * 2δ / P(X > 2)

Let X be an exponential random variable with parameter a.

a) The probability that X > 5 is given by the survival function of the exponential distribution,

which is P(X > 5) = e^(-a * 5).

b) The probability that X > 5 given that X > 2 is calculated using conditional probability.

The formula for conditional probability is P(X > 5 | X > 2) = P(X > 5 and X > 2) / P(X > 2).

Since X > 5 implies X > 2, the numerator is P(X > 5), which is e^(-a * 5). The denominator is P(X > 2), which is e^(-a * 2). Thus, P(X > 5 | X > 2) = (e^(-a * 5)) / (e^(-a * 2)).

c) Given that X > 2, and for a small δ > 0, the probability that 4 <= X <= 4 + 2δ is approximately

P(4 <= X <= 4 + 2δ | X > 2) = [P(4 <= X <= 4 + 2δ) - P(X < 2)] / P(X > 2).

We can approximate this by considering the probability density function (pdf) of the exponential distribution, which is f(x) = a * e^(-a * x).

The probability is approximately f(4) * 2δ / P(X > 2).

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The volume of the prism is determined as  103.0 unit³.

The volume of the triangular prism is calculated by applying the following formula as shown below;

V = ¹/₂bhl

where;

The base of the prism is calculated as follows;

tan 25 = 4/b

b = 4/tan (25)

b = 8.58 units

The volume of the prism is calculated as follows;

V = ¹/₂ x 8.58 x 6 x 4

V = 103.0 unit³

,

Thus, the volume of the prism is a function of its base, height and length.

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The area of the region of points satisfying the inequalities x ≤ 0, y ≤ 0, and y ≥ |x+4| - 5 is 4.5 square units.

if you graph the v shape on a graph, V , wiith vertex at (-4, -5) you can then make two triangles using the axis as a border.

The left triangle will have area 25/2

The right triangle witch will be smaller as it is below a rectangle will have area 8 and the rectangle will have area 4

Thus the total area is 49/2

To visualize the region of points satisfying the given inequalities, we can start by graphing the line y = |x+4| - 5.

That |x+4| is equal to x+4 when x is greater than or equal to -4, and -x-4 when x is less than -4.

Therefore, the equation of the line can be expressed as:

y = { x+9, for x ≤ -4 , -x-1, for x > -4

If you square both sides, then you get x+5 = 4[tex]x^2[/tex]

Which becomes polynomial 4[tex]x^2[/tex] -x -5

Factor to (4x-5)(x+1)

x = -1 and x = [tex]\frac{5}{4}[/tex]

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The mean and variance of X are 1.5 and 1. The mean and variance of Y = -X² - 2 are -5/2 and 41/8. The mean of W = sin(πY/4) is -1/2. The mean of Z = sin(X/4) is Σ sin(x/4).

a) The mean of a uniformly distributed random variable in the set {0, 1, 2, 3} is given by the formula:

mean = (a + b) / 2

where a and b are the lower and upper bounds of the distribution. In this case, a = 0 and b = 3, so:

mean = (0 + 3) / 2 = 1.5

The variance of a uniformly distributed random variable in the set {0, 1, 2, 3} is given by the formula:

variance = (b - a + 1)² / 12

So, in this case:

variance = (3 - 0 + 1)² / 12 = 1

b) Let Y = -X² - 2. We can use the properties of linear transformations of random variables to find the mean and variance of Y.

First, we find the mean of Y:

E(Y) = E(-X² - 2) = -E(X²) - 2

Next, we find the variance of Y:

Var(Y) = Var(-X² - 2) = Var(-X²) = E((-X²)²) - [E(-X²)]²

To find E((-X²)²), we need to calculate:

E((-X²)²) = E(X⁴) = Σ x⁴ P(X=x)

Since X is uniformly distributed in the set {0, 1, 2, 3}, we have:

E(X⁴) = (0⁴ + 1⁴ + 2⁴ + 3⁴) / 4 = 27/2

So,

Var(Y) = E(X⁴) - [E(X²)]² - 2 = 27/2 - (5/4)² - 2 = 41/8

Therefore, the mean of Y is -5/2, and the variance of Y is 41/8.

c) Let W = sin(πY/4). We can use the properties of linear transformations of random variables to find the mean of W.

E(W) = E(sin(πY/4)) = Σ sin(πy/4) P(Y=y)

We can find P(Y=y) by using the fact that X is uniformly distributed in the set {0, 1, 2, 3} and Y = -X² - 2:

P(Y=-2) = P(X=0) = 1/4

P(Y=-3) = P(X=1) = 1/4

P(Y=-6) = P(X=2) = 1/4

P(Y=-11) = P(X=3) = 1/4

So,

E(W) = sin(-π/2) (1/4) + sin(-3π/4) (1/4) + sin(-3π/2) (1/4) + sin(-11π/4) (1/4)

    = -1/4 - sqrt(2)/4 - 1/4 + sqrt(2)/4

    = -1/2

Therefore, the mean of W is -1/2.

d) Let Z = sin(X/4). We can use the properties of a uniformly distributed random variable to find the mean of Z.

E(Z) = E(sin(X/4)) = Σ sin(x/4)

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To create a 4x4 elementary matrix E that performs the row operation R2 - 3R1 → R2, you can follow this structure:
E = [1, 0, 0, 0]

     [-3, 1, 0, 0]
     [0, 0, 1, 0]
     [0, 0, 0, 1]

The 4 × 4 elementary matrix E that will carry out the row operation R2-3R, → R2 is:
1 0 0 0
-3 1 0 0
0 0 1 0
0 0 0 1

In this matrix, the entry in the second row and the first column is -3 because we are subtracting 3 times the first row from the second row. The other entries on the diagonal are 1 because we are not scaling those rows. The other entries in the second row are 0 because we are not adding or subtracting anything from those rows. The other entries in the matrix are also 0 because we are not modifying those rows. This matrix will perform the desired row operation when multiplied on the left of the original matrix.

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The maximum profit given the following revenue and cost functions is $12,000.

In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. In other words, a function takes an input, performs a certain operation on it, and produces a unique output. Functions are used to describe various real-world phenomena, and they are an essential tool in many branches of mathematics, science, and engineering.

Here,

To find the maximum profit, we need to first find the profit function which is given by:

P(x) = R(x) - C(x)

P(x) = (116x - x²) - (x³ - 6x² + 92x + 36)

P(x) = -x³ + x² + 24x - 36

To find the maximum profit, we need to take the derivative of P(x) and set it equal to zero:

P'(x) = -3x² + 2x + 24

-3x² + 2x + 24 = 0

Solving this quadratic equation gives:

x = 4 or x = -2/3

Since x represents the number of thousands of units produced, we reject the negative value and conclude that x = 4.

Therefore, the maximum profit is:

P(4) = -(4)³ + (4)² + 24(4) - 36

P(4) = -64 + 16 + 96 - 36

P(4) = $12,000 (in thousands of dollars)

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The expectation of Y is given by:

E[Y] = aVar(X) + (aE[X]^2 + bE[X] + c)

To derive the expectation of Y, we have:

E[Y] = E[ax^2 + bX + c]

Using the linearity of expectation, we can write:

E[Y] = E[ax^2] + E[bX] + E[c]

We know that E[c] = c, since the expected value of a constant is the constant itself. Also, E[bX] = bE[X], since b is a constant and can be taken outside the expectation operator. Therefore, we have:

E[Y] = aE[x^2] + bE[X] + c

To find E[x^2], we can use the fact that:

E[g(x)] = ∑ g(x) p(x)

Therefore, we have:

E[x^2] = ∑ x^2 p(x)

Since we don't know the specific distribution of X, we cannot calculate this directly. However, we can use a different formula for the variance of X, which is:

Var(X) = E[X^2] - E[X]^2

Rearranging this, we get:

E[X^2] = Var(X) + E[X]^2

Therefore, we can substitute this into our expression for E[Y], giving:

E[Y] = a(Var(X) + E[X]^2) + bE[X] + c

Simplifying this expression, we get:

E[Y] = aVar(X) + (aE[X]^2 + bE[X] + c)

Therefore, the expectation of Y is given by:

E[Y] = aVar(X) + (aE[X]^2 + bE[X] + c)

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∂w/∂r=-4r*cos(θ)/√(49-r²)

∂w/∂θ =0

After converting w to a function of r and θ, ∂w/∂r =-r/√(49-r²)

∂w/∂θ =0

(a) Using the chain rule, we have:

∂w/∂r = ∂w/∂x * ∂x/∂r + ∂w/∂y * ∂y/∂r

= (-4x/√(49-4x²-4y²)) * cos(θ) + (-4y/√(49-4x²-4y²)) * sin(θ)

= -4r*cos(θ)/√(49-r²)

Similarly,

∂w/∂θ = ∂w/∂x * ∂x/∂θ + ∂w/∂y * ∂y/∂θ

= (-4x/√(49-4x²-4y²)) * (-rsin(θ)) + (-4y/√(49-4x²-4y²)) * (rcos(θ))

= 0

Therefore, ∂w/∂r = -4r*cos(θ)/√(49-r²) and ∂w/∂θ = 0.

(b) Converting w to a function of r and θ, we have:

w = √(49 - 4r²(cos²(θ) + sin^2(θ)))

= 7√(1 - r²/7²)

Now, we can use the chain rule to find the partial derivatives:

∂w/∂r = (7/2)(1 - r²/7²)^(-1/2) * (-2r/7)

= -r/√(49-r²)

∂w/∂θ = (7/2)[tex]([/tex]1 - r²/7²[tex])^(^-^1^/^2^)[/tex] * 0

= 0

Therefore, ∂w/∂r = -r/√(49-r²) and ∂w/∂θ = 0, which are the same as the results obtained in part(a).

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The given series [tex]\sigma_n=1^\infty 5/n^3[/tex] is convergent.

The series given is:

Σₙ= [tex]1^\infty[/tex] 5/n³

The nth partial sum of this series is given by:

[tex]S_n = \sigma_k=1^n 5/k^3[/tex]

To calculate the first eight terms of the sequence of partial sums, we substitute n = 1, 2, 3, ..., 8 in the expression for Sₙ:

Rounding each partial sum to four decimal places, we get:

Based on these partial sums, it appears that the series is convergent. As we compute more and more terms of the sequence of partial sums, we observe that the sums increase, but at a decreasing rate, which suggests convergence.

To show that the series is convergent, we need to show that the sequence of partial sums approaches a finite limit as n approaches infinity.

We can use the Integral Test to show that the series converges. According to the Integral Test, if the series [tex]\sigma_n=1^\infty a_n[/tex]  is a series of non-negative terms and the integral from 1 to infinity of a continuous, positive, decreasing function f(x) is finite, then the series converges.

In this case, we can use f(x) = 5/x³ as the function and integrate from 1 to infinity:

Integral from 1 to infinity of 5/x³ dx = [5/(-2x²)] from 1 to infinity

= [tex]-5/2 \lim (x- > \infty)[1/x^2 - 1/1][/tex]

= 5/2

Since the integral of f(x) is finite, the series [tex]\sigma_n=1^\infty 5/n^3[/tex] converges by the Integral Test.

Therefore, the given series is convergent, as observed from the partial sums, and the sum of the series can be found by taking the limit of the sequence of partial sums as n approaches infinity.

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The problem has d)8 constraints (not counting the non-negativity constraints).

The problem is about determining the optimal production quantities for two products, in two months, in order to maximize revenue. The available time on each machine is 600 hours per month. The demands and prices for each product in each month are given in the problem.

To maximize revenue, we need to determine the quantity of each product to produce in each month, based on the demand and price constraints. We can write the objective function as:

Maximize: 60x₁₁ + 15x₁₂ + 70x₂₁ + 35x₂₂

where x₁₁ and x₁₂ are the quantities of product 1 produced in month 1 and month 2 respectively, and x₂₁ and x₂₂ are the quantities of product 2 produced in month 1 and month 2 respectively.

To ensure that we meet the demand for each product in each month, we have the following constraints:

x₁₁ + x₁₂ ≤ 120 (demand for product 1 in month 1 and 2)

x₂₁ + x₂₂ ≤ 150 (demand for product 2 in month 1 and 2)

x₁₁ ≤ 600 (available time on machine in month 1 for product 1)

x₁₂ ≤ 600 (available time on machine in month 2 for product 1)

x₂₁ ≤ 600 (available time on machine in month 1 for product 2)

x₂₂ ≤ 600 (available time on machine in month 2 for product 2)

To ensure that we do not produce negative quantities, we have the non-negativity constraints:

x₁₁ ≥ 0, x₁₂ ≥ 0, x₂₁ ≥ 0, x₂₂ ≥ 0

Therefore, the problem has a total of d)8 constraints (not counting the non-negativity constraints).

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Value of the iterated integral is 64.

To make it clearer, I'll rewrite the integral using proper notation:

∫(from 0 to 2) ∫(from 0 to 2x) ∫(from 0 to y) 3xyz dz dy dx

To evaluate the iterated integral, follow these steps:

1. Evaluate the innermost integral with respect to z:

∫(from 0 to 2) ∫(from 0 to 2x) [(3xyz²)/2] (from 0 to y) dy dx

2. Plug in the limits of integration for z:

∫(from 0 to 2) ∫(from 0 to 2x) [(3xy³)/2 - 0] dy dx

3. Evaluate the next integral with respect to y:

∫(from 0 to 2) [(3x²y⁴)/8] (from 0 to 2x) dx

4. Plug in the limits of integration for y:

∫(from 0 to 2) [(3x²(2x)⁴)/8 - 0] dx

5. Simplify the expression:

∫(from 0 to 2) [(3x¹⁰)/8] dx

6. Evaluate the outermost integral with respect to x:

[(3x¹¹)/88] (from 0 to 2)

7. Plug in the limits of integration for x:

[(3(2)¹¹)/88 - (3(0)¹¹)/88]

8. Simplify the expression:

(3 * 2048) / 88 = 6144 / 88 = 64

So the value of the iterated integral is 64.

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(a) The of constant c is: 1/15.

(b) The of constant c is: 1/10.

(c) The of constant c is: 1/36.

(d) The of constant c is: 1/2.

(a) We need to find the value of c such that f(x, y) satisfies the following properties:

f(x, y) >= 0 for all x and y

[tex]\sigma_x \sigma_y f(x, y) = 1[/tex], where the sums are taken over all possible values of x and y

Given f(x, y) = c(x + 2y), x = 1, 2, y = 1, 2, 3, we have:

[tex]\sigma_x \sigma_y f(x, y) = c(\sigma_x(x) + 2\sigma_y(y))[/tex]

= c((1+2+1)+(2+4+3))

= 15c

To satisfy property (2), we need:

15c = 1

Therefore, c = 1/15, and f(x, y) = (x+2y)/15 is the joint probability mass functions (pmf) for X and Y.

(b) We have f(x, y) = c(x + y), x = 1, 2, 3, y = 1, ..., x. Using the same reasoning as in part (a), we have:

[tex]\sigma_x \sigma_y f(x, y) = c(\sigma_x(x) + \sigma_x(x-1) + \sigma_x(x-2))[/tex]

= c(6+3+1)

= 10c

To satisfy property (2), we need:

10c = 1

Therefore, c = 1/10, and f(x, y) = (x+y)/10 is the joint pmf for X and Y.

(c) We have f(x, y) = c, where x and y are integers such that 9 <= x+y <= 18, 0 <= y <= 5. Using the same reasoning as in parts (a) and (b), we have:

[tex]\sigma_x \sigma_y f(x, y) = \sigma_x \sigma_y c[/tex]

[tex]= c \sigma_x \sigma_y 1[/tex]

= c (6)(6)

= 36c

To satisfy property (2), we need:

36c = 1

Therefore, c = 1/36, and f(x, y) = 1/36 is the joint pmf for X and Y.

(d) We have [tex]f(x, y) = c(1/4)^x (1/3)^y, x = 1, 2, ..., y = 1, 2, ....[/tex] Using the same reasoning as in parts (a), (b), and (c), we have:

[tex]\sigma_x \sigma_y f(x, y) = c \sigma_x ((1/4)^x) \sigma_y ((1/3)^y)[/tex]

= c (1/(1-(1/4))) (1/(1-(1/3)))

= c(4/3)(3/2)

= 2c

To satisfy property (2), we need:

2c = 1

Therefore, c = 1/2, and [tex]f(x, y) = (1/2)(1/4)^x (1/3)^y[/tex]is the joint pmf for X and Y.

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