The Pareto distribution with parameter 0 > 0 has a pdf as follows: f(x|0) = 0/x^0+1 0 x > 1 otherwise 。 Suppose the data: 5, 10, 8 was drawn independently from such a distribution. Find the maximum-likelihood estimate of 0.

Answers

Answer 1

The maximum likelihood estimate of θ for the given data is 1/3.

The likelihood function L(θ|x) for a sample of n observations x1, x2, ..., xn from a Pareto distribution with parameter θ is given by:

L(θ|x) = f(x1|θ) × f(x2|θ) × ... × f(xn|θ)

where f(xi|θ) is the probability density function of the Pareto distribution with parameter θ evaluated at xi.

Substituting the given pdf of the Pareto distribution with parameter 0, we get:

L(θ|x) = (θ/5θ) × (θ/10θ) × (θ/8θ) = θ³ / 4000

Taking the natural logarithm of the likelihood function, we get:

ln L(θ|x) = 3 ln θ - ln 4000

To find the maximum likelihood estimate (MLE) of θ, we differentiate ln L(θ|x) with respect to θ and set the derivative equal to zero:

d/dθ ln L(θ|x) = 3/θ = 0

Solving for θ, we get:

θ = 1/3

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Answer 2

The maximum likelihood estimate of θ for the given data is 0.501

Calculating the maximum likelihood of θ

From the question, we have the following parameters that can be used in our computation:

[tex]f(x|\theta) = \frac{\theta}{x^{\theta+ 1} }[/tex]

The likelihood function L(θ|x) for a Pareto distribution with parameter θ is calculated using

L(θ|x) = f(x₁|θ) * f(x₂|θ) * .....

Recall that

[tex]f(x|\theta) = \frac{\theta}{x^{\theta+ 1} }[/tex]

And

θ = 5, 10, 8

So, we have

[tex]L(\theta|x) = \frac{\theta}{5^{\theta+ 1} } * \frac{\theta}{8^{\theta+ 1} } * \frac{\theta}{10^{\theta+ 1} }[/tex]

Taking the natural logarithm both sides

[tex]\ln(L(\theta|x)) = \ln(\frac{\theta}{5^{\theta+ 1} } * \frac{\theta}{8^{\theta+ 1} } * \frac{\theta}{10^{\theta+ 1} })[/tex]

Differentiate

ln L'(θ|x) = -[(ln(10) + ln(8) + ln(5))θ - 3]/θ

Set the differentiated equation to 0

So, we have

-[(ln(10) + ln(8) + ln(5))θ - 3]/θ = 0

Solve for θ, we get:

(ln(10) + ln(8) + ln(5))θ = 3

So, we have

θ = 3/(ln(10) + ln(8) + ln(5))

Evaluate

θ = 0.501

Hence, the maximum likelihood of θ is 0.501

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Related Questions

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6 hrs x 60 min = 360 min

1170/360=3.25 laps/min

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Which figure has no line of symmetry

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Answer:

the first figure

Step-by-step explanation:

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Answer:

yes

Step-by-step explanation:

your m is your slope,your b is the y-intercept, and your y is what you are solving for.

Answer:

Y=mx+b is an equation for a linear equation

M is the slope of the line and B is the y-intercept

The y-intercept of this line is the value of y at the point where the line crosses the y axis.

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Is this a, b, c, or d?

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Answer:

Alright so let's start by elimination.

Step-by-step explanation:

It cannot be 1/4 (unless it's y or x)

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Answer:

2r+1+(-4r)+7

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Step-by-step explanation:

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Answers

Answer:  D) 1

=======================================================

Explanation:

The -2 outside the first matrix multiplies to every item inside that first matrix.

So the x updates to -2x in the upper left corner.

That -2x adds to the 3 in the upper left corner of the second matrix. The sum -2x+3 must be equal to the value in the upper left corner of the third matrix.

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Answer:

B

Step-by-step explanation:

did the math for it!!

B, jake.

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Find the value of x when the triangle has 9 and 3

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Answer:

H²=h²+b²

H²=9²+3²

H²=81+9

H²=90

H=

[tex] \sqrt{90} [/tex]

Gambarlah sebuah lingkaran dengan jari jari 2,5 cm. Lalu gambarlah 3 buah sudut keliling yg menghadap busur yang sama

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Step-by-step explanation:

isidhxibdofeixeosgzifisvsoshzibdihddgd

Madison was working at a store in the mall. She earns 7% commission for each sale that she makes. Last month, she sold a total of $1,450. How much commission did she earn?

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Answer:

101.50

Step-by-step explanation:

Answer:

$101.50

Step-by-step explanation:

7% of 1450 = 0.07 × 1450 = 101.5

Every year Ethan ears 38,428. Every year he spends 21,728 on expenses. How much money should Ethan have leftover?

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Answer:

16,700

Step-by-step explanation:

38,428

_

21,728

=

16,700

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Rosa is selling juice and cookies for a fundraiser. She earns $2 for every cookie and $3 for every bottle of juice. She wants to earn at least $24 from all the sales. Let x represent the number of cookies sold and y represent the number of bottles of juice sold. What does the slope of this inequality represent? 2x + 3y ≥ 24

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Answer:

The slope is 2. It represents the price of the cookies.

Step-by-step explanation:

The slope is 2 because its in frount of the x and 2 is the price of the cookies.

I don’t know if this is correct !!!!! Please help answer correctly !!!!!!!!! Will mark Brianliest !!!!!!!!!!!

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Answer:

change up to down the rest is right

Step-by-step explanation:

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Answer:

c = 4,9

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Sub -3 in as x:
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Answer:

1120 super easy

Step-by-step explanation:

8x10=80

80x14=1120

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Answers

Answer:  -15

Step-by-step explanation:

To find the y intercept we need to find slope first which can be done by taking 2 points and using the slope formula

5 - 10

----------  =   -5/3    

12--9

The slope is -5/3 now we take one point substitute the slope in and find the y intercept

5=-12( -5/3) +b

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The y-intercept is -15

Jane made a poster that had a length of 1 2/3 feet and a width of 5/6 feet. What is the perimeter of her poster?
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Answer:

5 feet

Step-by-step explanation:

At the student store, pencils are sold in either packs of three or packs of eight.
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Which numbers are impossible?

Answers

Answer

4, 7, 5, 10, 13, .......

Step-by-step explanation:

the only numbers that are impossible to get are the non-multiples of 3 and 8

and the numbers that 8 and 3 do not add up to and the multiples of 8 and 3 do not add up to

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Answer:

125

Step-by-step explanation:

Answer:

125

Step-by-step explanation:

368 - 243 = 125

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Answer:

There are many decimals between 2.10 and 2.11.

Examples:

2.101, 2.102, 2.103, 2.104, 2.105, 2.106, 2.107, 2.108, 2.109

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Answer:

2.25

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Answers

Answer:

39/30= 13/10

Step-by-step explanation:

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Proving trapezoid theorems Geometry

Answers

Answer:

ΔABD ≅ ΔACD by SAS, therefore;

[tex]\overline {BD} \cong \overline {CA}[/tex] by CPCTC

Step-by-step explanation:

The two column proof is presented as follows;

Statement [tex]{}[/tex]                   Reason

ABCD is a trapezoid [tex]{}[/tex]  Given

[tex]\overline {BA} \cong \overline {CD}[/tex]  [tex]{}[/tex]                   Given

[tex]\overline {BC} \parallel \overline {AD}[/tex]   [tex]{}[/tex]                   Definition of a trapezoid

ABCD is an isosceles trapezoid  [tex]{}[/tex]    Left and right leg are equal

∠BAD ≅ ∠CDA  [tex]{}[/tex]          Base angle of an isosceles trapezoid are congruent

[tex]\overline {AD} \cong \overline {AD}[/tex]   [tex]{}[/tex]                  Reflexive property

ΔABD ≅ ΔACD   [tex]{}[/tex]         By SAS rule of congruency

[tex]\overline {BD} \cong \overline {CA}[/tex]   [tex]{}[/tex]                  CPCTC

CPCTC; Congruent Parts of Congruent Triangles are Congruent

SAS; Side Angle Side rule of congruency

Answer:

Step-by-step explanation:

just did it

Which of these graphs represents a function?
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-5-4-3-2-11
-5
-5
w.
X.
.4-3-2-1,
1 2 3
5-4-3-2-13
1 2 3 4 5
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Y.

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Since each value of the input is mapped to only one value of the output, we have that graph w represents a function.

When does a graph represents a function?

When it has no vertically aligned points, that is, each value of x is mapped to only one value of y.

Hence, in this problem, graph w represents a function.

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Answer:

it's A have a good day

Step-by-step explanation:

You can upgrade lighting at your factory to LED bulbs that coast 6.95 each and last an average of 5 years.

Answers

Answer:

$200

Step-by-step explanation:

Here is the full question

You can upgrade lighting at your factory to LED bulbs that cost $6.95 each and last an average of 5 years. It costs $3 in labor to change a bulb. Over a 10-year period, about how much will it cost per year to install LED bulbs in 100 lamps and change the bulbs when they burn out?

in 10 years, the bulbs would be changed twice

Total cost incurred per bulb in 10 years = (6.95 +  $3) x 2 = $19.90

Since there are 100 bulbs, total cost incurred would be - $19.90 x 100 = $1990

Cost per year = $1990 /  (6.95 +  $3) = $200

Find the measure of each angle in the angle pair described.
The measure of one angle is twice the measure of its supplement. What are the measures of the two angles?

Answers

Answer:

The best way to measure an angle is to use a protractor. To do this, you'll start by lining up one ray along the 0-degree line on the protractor. Then, line up the vertex with the midpoint of the protractor. Follow the second ray to determine the angle's measurement to the nearest degree

You did not attach a photo of the angles!

The hypotenuse of a right triangle measures 15 cm and one of its legs
measures 5 cm. Find the measure of the other leg. If necessary, round to the
nearest tenth.

Answers

Answer:

[see below]

Step-by-step explanation:

Use the Pythagorean Theorem.

[tex]a^2+b^2=c^2[/tex]

'c' will be the hypotenuse.

The hypotenuse is 15 cm.  One of the legs is 5 cm.

[tex]5^2+b^2=15^2\\\\25 + b^2 = 225\\\\25-25+b^2=225-25\\\\b^2=200\\\\\sqrt{b^2}=\sqrt{200}\\\\ b=14.1421356237..., -14.1421356237[/tex]

14.1421356237 ≈ 14.1

The other leg should be about 14.1 cm.

Hope this helps.

The length of the other leg is 14.14 cm rounded to two decimal places.

What is Pythagoras's theorem?

In a right-angled triangle, the sum of the squares of the smaller two sides of a right-angle triangle is equal to the square of the largest side.

Given, The hypotenuse of a right triangle measures 15 cm, and one of its legs measures 5 cm.

We know,

Hypotenuse² = Base² + Adjacent².

Therefore,

15² = 5² + Adjacent².

Adjacent = √(225 - 25).

Adjacent = √200.

Adjacent = 14.14.

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Rahul sold a painting at 10% profit. If he had sold it for 240 more, he would have made 15% profit. What is the C.P. of the painting?

please tell ans step by step..

Answers

Answer:

$4800

Step-by-step explanation:

Let x be the C.P of the painting.

Profit=10% of x

Profit=[tex]\frac{10}{100}\times x=\frac{10x}{100}[/tex]

If S.P increase by 240 then

Profit=15% of x=[tex]\frac{15x}{100}[/tex]

For first case

S.P=C.P+ Profit

S.P=x+10x/100

S.P=[tex]\frac{100x+10x}{100}=\frac{110}{100}x[/tex]

For second case

New S.P[tex]=x+\frac{15}{100}x[/tex]

         =[tex]\frac{115}{100}x[/tex]

According to question

[tex]240+S.P=\frac{115}{100}x[/tex]

[tex]240+\frac{110x}{100}=\frac{115}{100}x[/tex]

[tex]240=\frac{115}{100}x-\frac{110x}{100}[/tex]

[tex]240=\frac{115x-110x}{100}[/tex]

[tex]240=\frac{5}{100}x[/tex]

[tex]x=\frac{240\times 100}{5}[/tex]

[tex]x=4800[/tex]

Hence, the C.P of the painting=$4800

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