Let X be a single observation from the beta(θ, 1) pdf.
(a) Let Y­ = −(log X)−1. Evaluate the confidence coefficient of the set [y/2, y].
(b) Find a pivotal quantity and use it to set up a confidence interval having the same confidence coefficient as the interval in part (a).
(c) Compare the two confidence intervals.

Answers

Answer 1

They both have the same confidence coefficient of 1/2, meaning that they both have a 50% chance of containing the true parameter value. Ultimately, the choice between the two intervals would depend on the specific goals of the analysis and the trade-offs between precision and coverage.

(a) We have that X ~ Beta(θ,1) and Y = -(log X)^-1. We need to find the confidence coefficient of the set [Y/2, Y]. Since Y is a transformation of X, we can use the transformation theorem to find the distribution of Y:

Let g(x) = -(log x)^-1. Then g'(x) = (1/x)(log(x)^-2), and so by the transformation theorem, we have that Y ~ Beta(1,θ).

Now we can use the properties of the Beta distribution to find the confidence coefficient of [Y/2, Y]:

P(Y/2 ≤ Y ≤ Y) = P(1/2 ≤ X ≤ 1) = Beta(θ,1)(1) - Beta(θ,1)(1/2) = 1/2.

Therefore, the confidence coefficient of [Y/2, Y] is 1/2.

(b) To find a pivotal quantity, we can use the fact that if X ~ Beta(θ,1), then X/(1-X) ~ Beta(θ,1). Let Z = X/(1-X). Then we have:Z ~ Beta(θ,1)

log(Z) ~ log(Beta(θ,1))

log(Z) ~ Σ(log(X[i])) - (n+1)log(1-X[i])

Since Z is a pivotal quantity, we can use it to construct a confidence interval with the same confidence coefficient as [Y/2, Y]. We have:

P(Y/2 ≤ Y ≤ Y) = P(log(Y) ≥ -2log(2)) - P(log(Y) > -log(2))

= P(log(Z) ≤ 2log(2)) - P(log(Z) > log(2))

= 1 - 2B(θ,1)(2^(-2)) - B(θ,1)(2^(-1))

Therefore, a confidence interval with the same confidence coefficient as [Y/2, Y] is given by:[exp(-2log(2)), exp(-log(2))] = [1/4, 1/2]

(c) Comparing the two confidence intervals, we can see that they have different widths. The interval [Y/2, Y] has a width of Y/2, while the interval [1/4, 1/2] has a width of 1/4. The interval [Y/2, Y] is centered around Y, while the interval [1/4, 1/2] is centered around 3/8. Therefore, the two intervals provide different information about the location and spread of the distribution.

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Related Questions

the sum of 8 and B is greater than 22

Answers

Given:

The sum of 8 and B is greater than 22.

To find:

The inequality for the given statement and its solution.

Solution:

We know that, sum of two number is the addition of two numbers.

Sum of 8 and B = 8+B

It is given that, the sum of 8 and B is greater than 22.

[tex]8+B>22[/tex]

Subtracting 8 from both sides, we get

[tex]8+B-8>22-8[/tex]

[tex]B>14[/tex]

Therefore, the required inequality for the given statement is [tex]8+B>22[/tex] and the solution is [tex]B>14[/tex].

What is the prime factorization of 91?

Answers

Answer:

The factors of 91 by prime factorization are 1, 7, 13 and 91. Here, 7, and 13 are the prime factors of 91.

Step-by-step explanation:

What is the value of so that −10and 10are both solutions of 2−=70

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Answer:I think 7

Step-by-step explanation:

What is the operation used in 3r?
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Answer:

Multiplication

Step-by-step explanation:

3r means 3 times R. We don't write it just like we don't write 1.0 for 1 or 1.000... A number in front of a variable means that number times that variable.

Hope this helps! <3

Find the solution of y = 5x – 4 for x = –6.


(26, –6)

(-6, -[tex]\frac{2}{5}[/tex])

(–6, –50)

(–6, –34)

Answers

The answer is (-6,-34)
Because since x = -6 u substituted so the equation will be y= 5(-6)-4 so y=-34 making the answer (-6,-34)

The space that sand fills varies directly with its mass. If 2,400 kg of sand fill 1.5 cubic meters, how much sand is needed to fill 2.25 cubic meters? PLEASE I REALLY NEED HELP!

Answers

Answer:

3,600 kg

Step-by-step explanation:

2,400÷6=400

400×3=1200

2,400+1,200= 3,600 kg

what is the measure of EGF?​

Answers

Answer:

i think c

Step-by-step explanation:

Answer:

B)  40.18°

Step-by-step explanation:

since it's a right triangle and you know values for opposite and hypotenuse you can use:

sin x = 12/18.6

this comes out to be 40.18°

what is the inequality of -5x + 19 ≥ 9

Answers

Answer:

x<=2

Step-by-step explanation:

-5x+19>=9

-5x>=9-19

-5x>=-10

5x>=10

x>=10/5

x>=2

x<=2

Angela can paint 3 sections in 25 minutes.
How many minutes will it take Angela to
paint 12 sections?

Answers

Answer:

Análisis del problema

Nos dan como datos el tiempo que se tarda cada uno en pintar la habitación:

Gerardo: 3 horas

Ángel: 6 horas

La velocidad de trabajo podemos definirla como dada por la cantidad de habitaciones que pueden pintar sobre el tiempo: H/t

Gerardo: 1 hab /3h

Ángel: 1 hab / 6h

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R950 is divided between Ann, Jill, Agnes Ann gets R100 more than Jill, Jill gets R50 more than Agnes. How much did Agnes get

Answers

Answer:
250

Explanation:
Agnes = x
Jill = (x+50)
Ann = (x+150)

x+(x+50)+(x+150)=950
3x + 200 = 950
3x = 750
x = 250

Agnes = 250
Ann = 400
Jill = 300

5x + 2y = 10
What is the substitution?

Answers

Hi there!

The substitutions are 10 and 0.

Compounds A and B are used in an experiment.

The equation y=700(0.8)x represents the remaining amount of compound A, in grams, after x hours. The table shows the remaining amount of compound B.

Time (h) Grams of B
1 800
2 400
3 200
What is the positive difference in the initial amount of each compound?

Enter your answer in the box.

Answers

9514 1404 393

Answer:

  900 grams

Step-by-step explanation:

The equation tells you the initial amount of A is 700 grams.

The table tells you B is being cut in half each hour, so the amount when h=0 is 2×800 = 1600 grams.

The difference in the initial amounts is ...

  1600 -700 = 900 . . . grams

I know the answer for this isn't a , but I'm not sure what the answer is. Cuz I would think you would be z y , y x, and XZ but that's not answer choice ​

Answers

Answer:

its the first one

Step-by-step explanation:

How many pounds are equal to 400 ounces

Answers

Answer: 25 pounds

Step-by-step explanation:

1 pound = 16 oz

400 oz / 16 oz = 25 pounds

I am not a professional, I am simply using prior knowledge!

Note- It would mean the world to me if you could mark me brainliest!

In rectangle PQRS, SP = 3 and RS = 4. What is the length of PR? (Hint: use
the Pythagorean theorem to solve for the diagonal) *

Answers

The answer is B). 5.

Both triangles share the same hypotenuse, PR, so you can use the lengths of the legs of the other triangle to find the answer.
4^2 + 3^2 = x^2
25 = x^2
The square root of 25 is 5, so that’s the answer.

What is an equation of the line that passes through the points (2, -2) and (8,4)?
Please I’ll give you BRAINLEST

Answers

Answer:

y=x-4

Step-by-step explanation:

m=(y2-y1)/(x2-x1)=(4-(-2))/(8-2)=(4+2)/6=6/6=1

slope is 1

y-y1=m(x-x1) where m=slope

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y=x-2-2

y=x-4

Which ordered pair is in the solution set of the inequality shown?

Answers

Answer:

number 2: (-6, 0)

AVALIAÇÃO DIAGNÓSTICA
Resolva os problemas abaixo
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feijão?
2) O aluguel de uma casa teve um reajuste e passou de R$ 500,00 para R$ 925,00. Qual foi o percentual
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Answer:

hhhsyeuru isuegrjro ehgeurov didyrhy

Triangle D E F is shown. Angle D E F is a right angle. The length of hypotenuse DF is 13, the length of E F is 5, and the length of E D is 12.
What is the length of the side opposite ∠D?


units

sin(D) =

Answers

Answer:

5 and 5/13

Step-by-step explanation:

What is the length of the side opposite ∠D?

✔ 5

units

sin(D) =

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What number should be added to both sides of the equation to complete the square?

x^2-10x=12​

Answers

Answer:

Step-by-step explanation:

x²-10x = 12

Coefficient of the x term: -10

Divide in half: -5

Square it: (-5)² = 5²

Add 5² to both sides:

x²-10x+5² = 12+5²

(x-5)² = 37

any help would be cool

Answers

Step-by-step explanation:

the shape has been split into 2 shapes. one cuboid and a triangular prism . work out the volume of each shape .

hint : volume : h×l×w ( height times length times width)

so the cubiod is 12 times 29 times 12 =

you work out the answer for that .

the triangular prism you try working out the volume.

Steven sold his old car and deposited the $3500 in a savings account at 3.5 percent interest for 2.5 years.
What was the total amount of money in the account at the end of this time?

Answers

Answer:

3500.035

I'm not sure

.

Step-by-step explanation:

.

Answer: It’s $3,806.25

Step-by-step explanation:

No explanation, I did the quiz and the other person who answered gave an answer that wasn’t in the multiple answers so I guessed and got it correct.

if a dozen eggs cost $3.84 calculate the cost of 30 eggs​

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Answer:

30

Step-by-step explanation:

Answer the photo below thanks

Answers

Answer:

x=50

Step-by-step explanation:

65+65=130

180-130=50

Answer:

50 is the answer

Step-by-step explanation:

this is a isosceles triangle having 65 and 65

then 65 + 65 + x = 180 property of traingle

130 + x = 180

x = 180- 130 = 50

The Students' Conjectures Emily and Zach have two different polynomials to multiply: Polynomial product A: (4x2 – 4x)(x2 – 4) Polynomial product B: (x2 + x – 2)(4x2 – 8x)

Answers

Complete Question:

Emily and Zach have two different polynomials to multiply: Polynomial product A: (4x2 – 4x)(x2 – 4) Polynomial product B: (x2 + x – 2)(4x2 – 8x) They are trying to determine if the products of the two polynomials are the same. But they disagree about how to solve this problem.

Answer:

[tex]A = B[/tex]

Step-by-step explanation:

See comment for complete question

Given

[tex]A: (4x^2 - 4x)(x^2 - 4)[/tex]

[tex]B: (x^2 + x - 2)(4x^2 - 8x)[/tex]

Required

Determine how they can show if the products are the same or not

To do this, we simply factorize each polynomial

For, Polynomial A: We have:

[tex]A: (4x^2 - 4x)(x^2 - 4)[/tex]

Factor out 4x

[tex]A: 4x(x - 1)(x^2 - 4)[/tex]

Apply difference of two squares on x^2 - 4

[tex]A: 4x(x - 1)(x - 2)(x+2)[/tex]

For, Polynomial B: We have:

[tex]B: (x^2 + x - 2)(4x^2 - 8x)[/tex]

Expand x^2 + x - 2

[tex]B:(x^2 + 2x - x - 2)(4x^2- 8x)[/tex]

Factorize:

[tex]B:(x(x + 2) -1(x + 2))(4x^2- 8x)[/tex]

Factor out x + 2

[tex]B:(x -1) (x + 2)(4x^2- 8x)[/tex]

Factor out 4x

[tex]B:(x -1) (x + 2)4x(x- 2)[/tex]

Rearrange

[tex]B: 4x(x - 1)(x - 2)(x+2)[/tex]

The simplified expressions are:

[tex]A: 4x(x - 1)(x - 2)(x+2)[/tex] and  

[tex]B: 4x(x - 1)(x - 2)(x+2)[/tex]

Hence, both polynomials are equal

[tex]A = B[/tex]

Please help quick will give brainliest!

Answers

Answer:

The function's behavior close to the vertical asymptote, x = 2 is;

[tex]\lim\limits_{x \to 2^-} f(x) = \infty[/tex] and [tex]\lim\limits_{x \to 2^+} f(x) = -\infty[/tex]

Step-by-step explanation:

From the graph of the function. we have;

1) As 'x' approaches 2 from the left (x → 2⁻), the graph with 'u' shape located in the middle of the three graphs, approaches infinity (arrow pointing up)

2) As 'x' approaches 2 from the right (x → 2⁺), the graph with 'n' shape located at the right side of the three graphs, approaches negative infinity (-∞) (arrow pointing down)

Therefore, the function's behavior close to the vertical asymptote, x = 2 is given as follows;

[tex]\lim\limits_{x \to 2^-} f(x) = \infty[/tex] and [tex]\lim\limits_{x \to 2^+} f(x) = -\infty[/tex]

Consider the function f(x) =2x+3 Find the average rate of change from the following points x=2 and x= 6

Answers

Step-by-step explanation:

step 1. m = 2.

step 2. for a linear equation the average rate of change between any points is the slope m.

Write −(1/4⋅1/4⋅1/4⋅1/4) using exponents.

Answers

Answer:

-(1/4)^4

Step-by-step explanation:

Hope it helped!!!

The student council was preparing for the wellness sale. The members divided each type of the donated items equally onto plates. Each plate contained only 1 type of item and every plate had exactly the same number of items. There were no leftovers. What is the greatest number of items that could have been placed on each​ plate? Wellness Sale Donations Celery 63 Cucumbers 14 Carrots 35 The greatest number of items that each plate could have is nothing items. ​(Type a whole​ number.)

Answers

Answer:

The greatest number of items on each plate is 7

Step-by-step explanation:

Given

[tex]Celery = 63[/tex]

[tex]Cucumber = 14[/tex]

[tex]Carrot = 35[/tex]

Required

Determine the greatest number of items on each plate

To do this, we simply calculate the GCF of 63, 14 and 35

[tex]63 =3 * 3 * 7[/tex]

[tex]14 = 2 * 7[/tex]

[tex]35 = 5* 7[/tex]

The GCF is:

[tex]GCF = 7[/tex]

Hence:

The greatest number of items on each plate is 7

I need help on this one math problem

Answers

OK well tell me and I’ll try to solve for you but I’ll trying to teach it to

Answer:

B

I hope this helps.....

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