The text discusses a production function and addresses various aspects of a firm's decision-making. It covers profit maximization, factor demand functions, supply function, profit function, Hotelling's lemma, cost minimization, conditional factor demand functions, and the cost function. These concepts are derived using mathematical calculations and formulas. Hotelling's lemma is verified, and the cost function is determined.
(a) The firm's profit maximization problem can be stated as follows: Maximize profits (π) by choosing the optimal levels of inputs (z and zo) that maximize the output (y) given the prices of output (p) and inputs (w, w₂).
(b) To derive the firm's factor demand functions, we need to find the conditions that maximize profits.
The first-order condition for input z is given by:
∂π/∂z = p * (∂f/∂z) - w = 0
Substituting the production function f(z) = z^(1/4) / z^(1/2) into the above equation, we have:
p * (1/4 * z^(-3/4) / z^(1/2)) - w = 0
Simplifying, we get:
p * (1/4 * z^(-7/4)) - w = 0
Solving for z, we find:
z = (4w/p)^(4/7)
Similarly, for input zo, the first-order condition is:
∂π/∂zo = p * (∂f/∂zo) - w₂ = 0
Substituting the production function f(zo) = z^(1/4) / z^(1/2) into the above equation, we have:
p * (1/2 * z^(1/4) * zo^(-3/2)) - w₂ = 0
Simplifying, we get:
p * (1/2 * z^(1/4) * zo^(-3/2)) - w₂ = 0
Solving for zo, we find:
zo = (2w₂ / (pz^(1/4)))^(2/3)
(c) To derive the firm's supply function, we need to find the level of output (y) that maximizes profits.
Using the production function f(z), we can express y as a function of z:
y = z^(1/4) / z^(1/2)
Given the factor demand functions for z and zo, we can substitute them into the production function to obtain the supply function for y:
y = (4w/p)^(4/7)^(1/4) / (4w/p)^(4/7)^(1/2)
Simplifying, we get:
y = (4w/p)^(1/7)
(d) The firm's profit function is given by:
π = p * y - w * z - w₂ * zo
Substituting the expressions for y, z, and zo derived earlier, we have:
π = p * ((4w/p)^(1/7)) - w * ((4w/p)^(4/7)) - w₂ * ((2w₂ / (pz^(1/4)))^(2/3))
(e) To verify Hotelling's lemma, we need to calculate the partial derivatives of the profit function with respect to the prices of output (p), input z (z₁), and input zo (z₂).
Hotelling's lemma states that the partial derivatives of the profit function with respect to the prices are equal to the respective factor demands:
∂π/∂p = y - z * (∂y/∂z) - zo * (∂y/∂zo) = 0
∂π/∂z₁ = -w + p * (∂y/∂z₁) = 0
∂π/∂z₂ = -w₂ + p * (∂y/∂z₂) = 0
By calculating these partial derivatives and equating them to zero, we can verify Hotelling's lemma.
(f) The firm's cost minimization problem can be stated as follows: Minimize the cost of production (C) given the level of output (y), prices of inputs (w, w₂), and factor demand functions for inputs (z, zo).
(g) To derive the firm's conditional factor demand functions, we need to find the conditions that minimize costs. We can express the cost function as follows:
C = w * z + w₂ * zo
Taking the derivative of the cost function with respect to z and setting it to zero, we get:
∂C/∂z = w - p * (∂y/∂z) = 0
Simplifying, we have:
w = p * (1/4 * z^(-3/4) / z^(1/2))
Solving for z, we find the conditional factor demand for z.
Similarly, taking the derivative of the cost function with respect to zo and setting it to zero, we get:
∂C/∂zo = w₂ - p * (∂y/∂zo) = 0
Simplifying, we have:
w₂ = p * (1/2 * z^(1/4) * zo^(-3/2))
Solving for zo, we find the conditional factor demand for zo.
(h) The firm's cost function is given by:
C = w * z + w₂ * zo
Substituting the expressions for z and zo derived earlier, we have:
C = w * ((4w/p)^(4/7)) + w₂ * ((2w₂ / (pz^(1/4)))^(2/3))
This represents the firm's cost function.
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A number cube has sides numbered 1 through 6. The probability of rolling a 2 is 1/6. What is the probability of not rolling a 2?
a. 1/6
b. 5/6
c. 1/5
d. 1/4
Probability refers to the measure of the likelihood that a particular event will occur. It is represented as a value between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.
The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6.
Here's why: When we roll a number cube with sides numbered 1 through 6, there are six possible outcomes, each with an equal probability of 1/6:1, 2, 3, 4, 5, 6.The probability of rolling a 2 is 1/6, which means there is only one way to roll a 2 out of the six possible outcomes. The probability of not rolling a 2 is the probability of rolling any of the other five possible outcomes. Each of these outcomes has an equal probability of 1/6. Therefore, the probability of not rolling a 2 is:1 - (1/6) = 5/6. Answer: b. 5/6.
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Given that the number cube has sides numbered 1 through 6. The probability of rolling a 2 is 1/6. The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6.
The probability of rolling any of the numbers 1, 3, 4, 5, or 6 is also 1/6 each.
The sum of the probabilities of all possible outcomes is 1.
The probability of an event happening is defined as the number of ways the event can occur, divided by the total number of possible outcomes.
The total number of possible outcomes is 6 (the numbers 1 through 6).
Thus, if the probability of rolling a 2 is 1/6, then the probability of not rolling a 2 is 1 - 1/6 = 5/6.
Therefore, the correct option is b. 5/6.
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Select 2A316 in base 10.
(q17) A geologist finds out that a radioactive substance A that he found in the caves of Africa decays at a rate of 0.03 percent every year. What is the probability that an atom of this substance chosen at random will decay in the next 70 years?
None of the given options is the answer.
To calculate the probability of decay for substance A over the next 70 years, we need to consider the decay rate of 0.03 percent per year.
The decay rate of 0.03 percent per year can be converted to a decimal by dividing it by 100: 0.03 / 100 = 0.0003.
The probability of an atom decaying in a given year is equal to the decay rate, which is 0.0003.
To calculate the probability of an atom not decaying in a given year, we subtract the decay rate from 1: 1 - 0.0003 = 0.9997.
The probability of an atom not decaying over the next 70 years can be calculated by multiplying the probability of not decaying in each year together: (0.9997)^70 ≈ 0.9704.
Therefore, the probability of an atom decaying in the next 70 years is equal to 1 minus the probability of not decaying: 1 - 0.9704 ≈ 0.0296.
So, the probability that an atom of substance A chosen at random will decay in the next 70 years is approximately 0.0296 or 2.96%.
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For 25 pts
Pls Help this is hard as hell
Answer: For the first one Independent variable would be Cars age and the dependent would be cars price. For the second one, independent variable would be number of training miles and dependent would be Time to finish the race in minutes.
Step-by-step explanation:
Answer:
First one:
The independent variable is the car’s age
The dependent variable is the car’s price according tot he age
Second one:
The independent variable is the number of training miles
The dependent variable is the time it takes to finish
Step-by-step explanation:
Just think of the independent variable as the cause and the dependent variable as the effect.
What is the volume of the pyramid in
cubic centimeters?
Answer:
3328 cubic centimeters
Step-by-step explanation:
volume of pyramid equation:
V=(lwh)/3
V = (12·26·32) / 3
V =3328
Answer:
The answer is
[tex]9984 cm {}^{3} [/tex]
Step-by-step explanation:
The way i solved this was by using the formula to volume. I also am doing this but for me it is a bit easier. The simple formula is Width x Length x Height. Since i already have the numbers, it is easier to plug in the numbers
What is the value of Point C on the number line below?
A) 0.208
B) 0.28
C) 0.302
D) 0.32
Answer:
0.28
Step-by-step explanation:
All you need to do is count.
0.20, 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.30
C
Point C sits on the point 0.28.
HELP PLEASE I NEED HELP !
Answer:
G
Step-by-step explanation:
out of a total of 280 spinners as the overall.
3/40 were defective
280 * 3/40 = 21
Answer:
G
Step-by-step explanation:
For every 40 spinners 3 are defective
Divide amount made by 40 for numbers of groups of 40
280 ÷ 40 = 7 , then
7 × 3 = 21 ← likely defective spinners → G
From Hardcover Book, Marsden/Tromba, Vector Calculus, 6th ed., Section 2.1., # 40) Using polar coordinates, describe the level curves of the function defined by f (x, y) = - 2xy (22+y2) if (x, y) + (0,0) and f(0,0) = 0.
The level curves of the function f(x, y) = -2xy / (2^2 + y^2) in polar coordinates consist of lines θ = π/2 + kπ and θ = kπ, as well as the upper half and lower half of the unit circle depending on the sign of the function. These level curves represent the points (r, θ) where the function f(r, θ) is constant.
To describe the level curves of the function f(x, y) = -2xy / (2^2 + y^2), we can first express the function in terms of polar coordinates. Let's substitute x = r cos(θ) and y = r sin(θ) into the function:
f(r, θ) = -2(r cos(θ))(r sin(θ)) / (r^2 + (r sin(θ))^2)
Simplifying this expression, we get:
f(r, θ) = -2r^2 cos(θ) sin(θ) / (r^2 + r^2 sin^2(θ))
Now, we can further simplify this expression:
f(r, θ) = -2r^2 cos(θ) sin(θ) / (r^2(1 + sin^2(θ)))
f(r, θ) = -2 cos(θ) sin(θ) / (1 + sin^2(θ))
The level curves of this function represent the points (r, θ) in polar coordinates where f(r, θ) is constant. Let's consider a few cases:
1. When f(r, θ) = 0:
This occurs when -2 cos(θ) sin(θ) / (1 + sin^2(θ)) = 0. Since the numerator is zero, we have either cos(θ) = 0 or sin(θ) = 0. These correspond to the lines θ = π/2 + kπ and θ = kπ, where k is an integer.
2. When f(r, θ) > 0:
In this case, the numerator -2 cos(θ) sin(θ) is positive. For the denominator 1 + sin^2(θ) to be positive, sin^2(θ) must be positive. Therefore, the level curves lie in the regions where sin(θ) > 0, which corresponds to the upper half of the unit circle.
3. When f(r, θ) < 0:
Similar to the previous case, the level curves lie in the regions where sin(θ) < 0, which corresponds to the lower half of the unit circle.
In summary, the level curves of the function f(x, y) = -2xy / (2^2 + y^2) in polar coordinates consist of lines θ = π/2 + kπ and θ = kπ, as well as the upper half and lower half of the unit circle depending on the sign of the function.
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Tell which value of the variable is the solution of the equation 30 = 6w W = 3, 5, 6, 8??
Answer: w=5
Step-by-step explanation: Hope this help
Someone please help!!! will give brainliest!!!
Round your answer to the nearest hundredths, if necessary.
Find the surface area of the figure
Answer:161.56
Step-by-step explanation:
8 x5=40
8 x 7.07=56.56
1/2 x 5 x 5 x 2= 25
8 x 5=40
Add that all together
Simple word problem. 40 POINTS!!!!Thank you.
Answer:
$50
Step-by-step explanation:
Hello There!
We are given that for 1 hour of work 250 dollars is charged and for 3 hours of work 350 dollars is charged
This could also be represented in two points (1,250) and (3,350)
The question wants us to find the hourly charge rate (slope)
we can easily find the slope ( hourly charge rate ) by using the slope formula
[tex]slope=\frac{y_2-y_1}{x_2-x_1}[/tex]
we have our two points so all we need to do is plug in the values (remember y values go on top and x values go on the bottom.)
[tex]slope=\frac{350-250}{3-1} \\350-250=100\\3-1=2\\slope=\frac{100}{2} or50[/tex]
So we can conclude that the hourly charge rate is $50
increase 50$ by 15%
Can you say how to do it and answer?
Solve the system of equations using the substitution method. Show your work and be sure to include the solution to the system.
let f be a function with derivative given by f x ¢( ) = 3 x + 1. what is the length of the graph of y f = ( )x from x = 0 to x = 1.5 ?
If "f" is function with derivative as f'(x) = √(x³ + 1), then length of graph of y = f(x) from x = 0 to x = 1.5 is (b) 2.497.
To find the length of the graph of y = f(x) from x = 0 to x = 1.5, we use the arc-length formula for a function y = f(x):
Length = ∫ᵇₐ√(1 + [f'(x)]²) dx,
Given the derivative : f'(x) = √(x³ + 1), we substitute it into the arc-length formula:
Length = [tex]\int\limits^{1.5}_{0}[/tex] √(1 + (√(x³ + 1))²) dx,
Simplifying the expression inside the square root:
We get,
Length = [tex]\int\limits^{1.5}_{0}[/tex] √(1 + x³ + 1) dx
= [tex]\int\limits^{1.5}_{0}[/tex]√(x³ + 2) dx
= 2.497.
Therefore, the correct option is (b).
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The given question is incomplete, the complete question is
Let f be a function with derivative given by f'(x) = √(x³ + 1). What is the length of the graph of y = f(x) from x = 0 to x = 1.5?
(a) 4.266
(b) 2.497
(c) 2.278
(d) 1.976
Worth five points! it doesnt tell me if what answer is right but if i get 75% or up i will mark the first person who answered with an actual answer brainliest and i don't lie about brainliest!! Please no nonsense answers I just want help :(
Triangle DEF is an isosceles triangle with DE = EF, and mE =92°
What is mD?
A. 45
B. 44°
C. 88°
D. 90°
Answer:
Step-by-step explanation:
Answer:
44 is the awnser
Step-by-step explanation:
becuase if you were to look at the first persons work it is correct showing him solving the equasion and witch it is not 44
Seats in a theater are curved from the front row to the back. The front row has 10 chairs, the second has 16 and the third has 22, and so on.
A. Write a recursive rule for this series
B. Write an explicit rule for this series
C. Using the explicit formula, find the number of chairs in row 5
D. The auditorium can hold 17 rows of chairs. Write a sigma notation for this series, and then use either series formula to calculate how many chairs can fit in the auditorium
Answer:
The first term is 10.
The second term is 16
The third term is 22.
We can see that the first term plus 6, is:
10 + 6 = 16
Then the first term plus 6 is equal to the second term.
And the second term plus 6 is:
16 + 6 = 22
Then the second term plus 6 is equal to the third term.
A) As we already found, the recursive rule is:
Aₙ = Aₙ₋₁ + 6
B) The explicit rule is:
Aₙ = A₁ + (n - 1)*6
Such that A1 is the first term, in this case A₁ = 10
Then:
Aₙ = 10 + (n - 1)*6
C)
Now we want to find A₅, then:
A₅ = 10 + (5 - 1)*6 = 34
There are 34 chairs in row 5.
D)
Here we have 17 rows, then we can have 17 terms, this means that the total number of chairs will be:
C = A₀ + A₁ + ... + A₁₆
This summation can be written as:
∑ 10 + (n - 1)*6 such that n goes from 0 to 16.
The formula for the sum of the first N terms of a sum like this is:
S(N) = (N)*(A₁ + Aₙ)/2
Then the sum of the 17 rows gives:
S(17) = 17*(10 + (10 + (17 - 1)*6)/2 = 986 chairs.
There are total 986 chairs in the considered auditorium and there are 34 chairs in the fifth row.
The recursive rule for this series is: [tex]T_n = T_{n-1} + 6[/tex]The explicit rule for this series is: [tex]T_n = 6n + 4[/tex]What is recursive rule?A rule defined such that its definition includes itself.
Example: [tex]F(x) = F(x-1) + c[/tex] is one such recursive rule.
For this case, we're provided that:
Seats in rows are 10 in front, 16 in second, 22 in third, and so on.
10 , 16 , 22 , .....
16 - 10 = 6
22 - 16 = 6
...
So consecutive difference is 6
If we take [tex]T_i[/tex] as ith term of the series then:
[tex]T_2 - T_1 = 6\\T_3 - T_2 = 6\\T_4 - T_3 = 6 \\T_5 - T_4 = 6\\\cdots\\T_{n} - T_{n-1} = 6[/tex]
Thus, the recursive rule for the given series is [tex]T_{n} - T_{n-1} = 6[/tex] or [tex]T_n = T_{n-1} + 6[/tex]
From this recursive rule, we can deduce the explicit formula as:
[tex]T_n = T_{n-1} + 6\\T_n = T_{n-2} + 6 + 6\\\cdots\\T_n = T_{n-k} + k \times 6\\T_n = T_1 + 6(n-1)\\T_n = 10 + 6(n-1) \: \rm (as \: T_1 = 10)\\[/tex]
Thus, the explicit rule for this series is [tex]T_n = 10 + 6(n-1)[/tex]
For 5th row, putting n = 5 gives us:
[tex]T_n = 10 + 6(n-1) = 6n + 4\\T_5 = 6(5) + 4 = 34[/tex]
If the auditorium has 17 rows, then total chairs are:
[tex]T = T_1 + T_2 + \cdots + T_{17} = \sum_{n=1}^{17} T_n\\\\T = \sum_{n=1}^{17} (10 + 6(n-1))\\\\T = \sum_{n=1}^{17} (6n + 4)\\\\T = 6\sum_{n=1}^{17} n + \sum_{n=1}^{17}4 = 6\sum_{n=1}^{17} n + 4 \times 17\\\\T = 6\left( \dfrac{17(18)}{2}\right) + 68 = 918 + 68\\\\T = 986[/tex]
(it is because [tex]\sum_{k=1}^n k = 1 + 2 + \cdots + n = \dfrac{n(n+1)}{2}[/tex] )
Thus, there are total 986 chairs in the considered auditorium. There are 34 chairs in the fifth row. The recursive rule for this series is: [tex]T_n = T_{n-1} + 6[/tex] The explicit rule for this series is: [tex]T_n = 6n + 4[/tex].
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66666666 help me plz plz plz
Answer:
XY would also be 7 centimeters which is answer D.
Step-by-step explanation:
This is a parallelogram, meaning that the adjacent sides are congruent. As well, the triangles making up the figure are congruent, so it makes sense that XY would also equal 7 centimeters.
which of the following expressions is equivalent to -10?
a.-7 3
b.-3 - 7
c.3 - 7
d.7 - 3
The expression which is equivalent to -10 is the option b, -3 - 7.
Explanation:
We can use subtraction and addition of integers to get the value of the given expression. We can write the given expression as;
-3 - 7 = -10 (-3 - 7)
The addition of two negative integers will always give a negative integer. When we subtract a larger negative integer from a smaller negative integer, we will get a negative integer.
If we add -3 and -7 we will get -10. This makes the option b the correct answer.
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PLEASE HELP !!!! find the focus (parabolas)
(y-2)^2=4(x+3)
Answer:
C. ( -2 , 2 )Step-by-step explanation:
Focus of parabola [tex](y-2)^2 = 4(x+3)[/tex] is (-2 , 2) .
Correct option is C .
Given, Equation of parabola [tex](y-2)^2 = 4(x+3)[/tex]
Focus of parabola :
Standard equation of parabola : (y - k)² = 4a(x - h)
Axis of parabola : y = k
Vertex of parabola : (h, k)
Focus of parabola : (h + a, k)
Compare the equation of parabola with standard equation.
(y - k)² = 4a(x - h)
[tex](y-2)^2 = 4(x+3)[/tex]
k = 2
a = 1
h = -3
So focus of parabola: (h + a, k).
-3 + 1 , 2
Focus of parabola = -2 , 2
Hence the correct option is C .
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the radius of a circle is 8 miles. what is the area of a sector bounded by a 144° arc
Answer:
Step-by-step explanation:
The area of a sector and the properties of circles bounded by a 144° arc in a circle with a radius of 8 miles can be calculated using the formula: Area of sector = (θ/360°) * π * r² where θ is the central angle of the sector and r is the radius of the circle.
In this case, the central angle is 144° and the radius is 8 miles. Plugging these values into the formula, we get: Area of sector = (144°/360°) * π * (8 miles)². Simplifying the equation, we have: Area of sector = (0.4) * π * (8 miles)².
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Which of the following sets of angle measures can be used to draw an acute isosceles triangle? Select all that apply. 75°, 30°, 75° 80°, 55°, 45° 80°, 80°, 40° 60°, 60°, 60° 50°, 50°, 80° 20°, 140°, 20°
Answer:
1. 80°, 80°, 40°
2. 60°, 60°, 60°
3. 20°, 140°, 20°
4. 50°, 50°, 80°
5. 75°, 30°, 75°
6. 80°, 55°, 45°
6, 2, and 4
Answer:
I think its 6,2, and 4
Step-by-step explanation: hope that helps! °∪°
y=5x
y=-3x+24
Solve by substitution
Answer:
x=12
Step-by-step explanation:
5x = -3x+24
2x = 24
x = 12
2, 3, 1, 6, 4, 5, 3, 2, 3, 4 is the set
Answer:
mean: 3.3
median: 3
mode: 3
range: 5
Q1 = 2
Q3 = 4
IQR = 2
Step-by-step explanation:
A rectangular swimming pool is 6 it deep. One side of the pool is 3.5 times longer than the other. The amount of water needed to fill the swimming pool is
1344 cubic feet. Find the dimensions of the pool.
Answer:
8 feet by 28 feet by 6 feet
Step-by-step explanation:
So volume is length times width times height
It tells us that the volume is 1344 cubic feet (the water used to fill it)
And it also tells us that the height/depth (which are the same thing in this case) is 6ft
All we need now are length and width
We know that one of the sides is 3.5 times the other one. So we can just say length is x and width is 3.5x
So plugging that in, the equation becomes
[tex]3.5x*x*6=1344[/tex]
3.5 x times x is just 3.5x squared so
[tex]3.5x^2*6=1344[/tex]
divide both sides by 6
[tex]3.5x^2=244[/tex]
divide by 3.5
[tex]x^2 =64[/tex]
[tex]x=\sqrt{64}[/tex]
x = 8
So that means the one side is 8 feet long and the other side is 3.5 times that, which is 28 feet long.
So the dimensions of the pool are 8 feet by 28 feet by 6 feet
A survey was conducted that asked 1014 people how many books they had read in the past year. Results indicated that x = 12.7 books and s= 16.6 books. Construct a 90% confidence interval for the mean number of books people read. Interpret the interval Click the icon to view the table of critical t-values. Construct a 90% confidence interval for the mean number of books people read and interpret the result. Select the correct choice below and fill in the answer boxes to complete your choice (Use ascending order. Round to two decimal places as needed) OA. There is 90% confidence that the population mean number of books read is between __ and __. if repeated samples are taken, 90% of them will have a sample mean between __ and __. There is a 90% probability that the true mean number of books read is between __ and __ .
There is 90% confidence that the population mean number of books read is between 9.85 and 15.55. If repeated samples are taken, 90% of them will have a sample mean between 9.85 and 15.55. There is a 90% probability that the true mean number of books read is between 9.85 and 15.55.
What is the 90% confidence interval for the mean number of books read?The survey results indicate that the mean number of books read in the past year is estimated to be 12.7, with a standard deviation of 16.6. To construct a 90% confidence interval, we can use the t-distribution and the sample size of 1014. Using the critical t-values from the table, we calculate the margin of error by multiplying the standard error (s / √n) with the t-value. Adding and subtracting the margin of error from the sample mean gives us the lower and upper bounds of the confidence interval.
The confidence interval for the mean number of books read is calculated as 12.7 ± (t-value * 16.6 / [tex]\sqrt{1014}[/tex]), which simplifies to 12.7 ± 2.58. Therefore, the confidence interval is (9.85, 15.55).
In interpretation, this means that we can be 90% confident that the true mean number of books read in the population falls between 9.85 and 15.55. If we were to repeat the survey and take different samples, 90% of those samples would produce a mean number of books read within the range of 9.85 to 15.55. The confidence interval provides a range of values within which we can reasonably estimate the true population mean.
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Consider a sequence of i.i.d random variables X₁, X2,..., each with a discrete uniform distribution on the set {0, 1,2}. In other words, P(X = 0) = 1/3 = P(X₁ = 1) = P(X = 2), for each k. (a) Compute P(X₁ + X₂ ≤ 1). (b) Determine the mgf of X₁ along with its domain. n (c) Consider a sequence of sample averages, {X}, where X₁ = EX for n € N. Find k=1 the mgf of X, by also stating its domain. Hint. First describe the mgf of X, in terms of the mgf of Xk, and then use the mgf of X.
(a) To compute P(X₁ + X₂ ≤ 1), we can list out all the possible values of X₁ and X₂ that satisfy the inequality: X₁ + X₂ ≤ 10 + 0 = 0, which is impossible, so P(X₁ + X₂ ≤ 1) = P(X₁ = 0, X₂ = 0) + P(X₁ = 1, X₂ = 0) + P(X₁ = 0, X₂ = 1) = (1/3)² + (1/3)² + (1/3)² = 1/3.
(b) The moment generating function (mgf) of X₁ is given by:
M(t) = E(etX₁) = (1/3) et0 + (1/3) et1 + (1/3) et2 = (1/3) + (1/3) et + (1/3) e2t
The domain of M(t) is the set of all values of t for which E(etX₁) exists.
(c) Let X be the sample average of {Xk}, where Xk are i.i.d random variables with the same distribution as X₁.
Then, by the linearity of expectation and the definition of X₁, we have:
E(X) = E( (X₁ + X₂ + ... + Xn)/n ) = (E(X₁) + E(X₂) + ... + E(Xn))/n = (EX₁ + EX₂ + ... + EXn)/n = X₁ = 1
From part (b), we have the mgf of X₁ as M₁(t) = (1/3) + (1/3)et + (1/3)e2t.
Then, the mgf of X is given by the formula: M(t) = E(etX) = et (X₁ + X₂ + ... + Xn)/n) = E(etX₁/n) × E(etX₂/n) × ... × E(etXn/n) = (M₁(t/n)) ⁿ = [(1/3) + (1/3) et/n + (1/3) e2t/n] ⁿ
The domain of M(t) is the set of all values of t for which E(etX) exists.
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the circumfrence is 72 cm what is the length of the minor arc
Answer:
Should be 9 centimeters.
Step-by-step explanation:
find the shaded region of the figure below
Answer:
-x³ + 3x² - 14x + 12
Step-by-step explanation:
Area of outer rectangle = (x² + 3x - 4) * (2x - 3)
= (x² + 3x - 4) * 2x + (x² + 3x - 4) * (-3)
=x²*2x + 3x *2x - 4*2x + x² *(-3) + 3x *(-3) - 4*(-3)
=2x³ + 6x² - 8x - 3x² - 9x + 12
= 2x³ + 6x² - 3x² - 8x - 9x + 12 {Combine like terms}
= 2x³ + 3x² - 17x + 12
Area of inner rectangle = (x² - 1)* 3x
= x² *3x - 1*3x
= 3x³ - 3x
Area of shaded region = area of outer rectangle - area of inner rectangle
= 2x³ + 3x² - 17x + 12 - (3x³ - 3x)
= 2x³ + 3x² - 17x + 12 -3x³ + 3x
= 2x³ - 3x³ + 3x² - 17x + 3x + 12
= -x³ + 3x² - 14x + 12
Can someone please help me with math.
EQUAÇO
1. x + 5 - 25=x + 3x - 4
2. 1 - 2x = 3 - 2(x + 1)