A random variable x is said to belong to the one-parameter exponential family of distributions if its pdf can be written in the form: Síx;6)=exp[AO)B(x) + C(x)+D(0)] where A(O), DCO) are functions of the single parameter 0 (but not x) and B(x), C(x) are functions of (but not ). Write down the likelihood function, given a random sample X,, X2,...,x, from the distribution with pdf f(x;0). (2 Marks) (b) If the likelihood function can be expressed as the product of a function which depends on 0 and which depends on the data only through a statistic T(x,x2,...,x.) and a function that does not depend on 0, then it can be shown that T is a sufficient statistic for 0. Use this result to show that B(x) is a a sufficient statistic for 0 in the one-parameter exponential family of part (b). (3 Marks) c) If the sample consists of iid observations from the Uniform distribution on the interval (0,0), identify a sufficient statistic for 0.
Answers
(a) The likelihood function for a random sample X1, X2, ..., Xn from the distribution with pdf f(x;θ) is given by:
L(θ|x1, x2, ..., xn) = ∏i=1^n f(xi;θ)
For the one-parameter exponential family of distributions, the pdf is given by:
f(x;θ) = exp[A(θ)B(x) + C(x) + D(θ)]
Therefore, the likelihood function can be written as:
L(θ|x1, x2, ..., xn) = exp[∑i=1^n A(θ)B(xi) + ∑i=1^n C(xi) + nD(θ)]
(b) If the likelihood function can be expressed as the product of a function which depends on θ and which depends on the data only through a statistic T(x1, x2, ..., xn), and a function that does not depend on θ, then T is a sufficient statistic for θ.
In the one-parameter exponential family of distributions, we can write the likelihood function as:
L(θ|x1, x2, ..., xn) = exp[nA(θ)B(T) + nC(T) + nD(θ)]
where T = T(x1, x2, ..., xn) is a statistic that depends on the data only and not on θ.
Comparing this to the general form, we see thatthe function that depends on θ is exp[nA(θ)B(T) + nD(θ)], and the function that does not depend on θ is exp[nC(T)]. Therefore, T is a sufficient statistic for θ.
To show that B(x) is a sufficient statistic for θ in the one-parameter exponential family, we need to show that the likelihood function can be written in the form:
L(θ|x1, x2, ..., xn) = h(x1, x2, ..., xn)g(B(x1), B(x2), ..., B(xn);θ)
where h(x1, x2, ..., xn) is a function that does not depend on θ, and g(B(x1), B(x2), ..., B(xn);θ) is a function that depends on θ only through B(x1), B(x2), ..., B(xn).
Starting with the likelihood function from part (a):
L(θ|x1, x2, ..., xn) = exp[∑i=1^n A(θ)B(xi) + ∑i=1^n C(xi) + nD(θ)]
Let's define:
h(x1, x2, ..., xn) = exp[∑i=1^n C(xi)]
g(B(x1), B(x2), ..., B(xn);θ) = exp[∑i=1^n A(θ)B(xi) + nD(θ)]
Now we can rewrite the likelihood function as:
L(θ|x1, x2, ..., xn) = h(x1, x2, ..., xn)g(B(x1), B(x2), ..., B(xn);θ)
which shows that B(x1), B(x2), ..., B(xn) is a sufficient statistic for θ in the one-parameter exponential family.
(c) If the sample consists of iid observations from the Uniform distribution on the interval (0, θ), then the pdf of each observation is:
f(x;θ) = 1/θ for 0 < x < θ
The likelihood function for a random sample X1, X2, ..., Xn from this distribution is:
L(θ|x1, x2, ..., xn) = ∏i=1^n f(xi;θ) = (1/θ)^n for 0 < X1, X2, ..., Xn < θ
To find a sufficient statistic for θ, we need to express the likelihood function in the form:
L(θ|x1, x2, ..., xn) = h(x1, x2, ..., xn)g(T(x1, x2, ..., xn);θ)
where T(x1, x2, ..., xn) is a statistic that depends on the data only and not on θ.
Since the likelihood function only depends on the maximum value of the sample, we can define T(x1, x2, ..., xn) = max(X1, X2, ..., Xn) as the maximum of the observed values.
The likelihood function can then be written as:
L(θ|x1, x2, ..., xn) = (1/θ)^n * I(x1, x2, ..., xn ≤ θ)
where I(x1, x2, ..., xn ≤ θ) is the indicator function that equals 1 if all the observed values are less than or equal to θ, and 0 otherwise.
We can see that the likelihood function depends on θ only through the term 1/θ, and the function I(x1, x2, ..., xn ≤ θ) depends on the data only and not on θ. Therefore, T(x1, x2, ..., xn) = max(X1, X2, ..., Xn) is a sufficient statistic for θ in the Uniform distribution on the interval (0, θ).
Related Questions
PLSSS HELPPPPP
On the following number line, point C represents the integer -1. Identify the integer that each of the other letters represent.
A:
B:
D:
E:
Answers
Answer: B is 0 A is 1 D 2 E 3
Step-by-step explanation:
Answer:
a: 1
b: 0
d:2
e:3
Step-by-step explanation:
Since you already know that C is -1, you can add or subtract along the numberline by ones to get the values of each letter
Is the line y = 0.6x - 1
Answers
Answer:yes!
Step-by-step explanation:
Answer:
Yes
Step-by-step explanation:
To prove check out this screenshot Mark my answer the brainliest to tell you the 4 math calculators :)
x - 3.5 = -3.1
solve for x.
Answers
Answer:
0.4
Step-by-step explanation:
The numbers -3.5 and -3.1 have a difference in value of 0.4
An easy way to look at this is to remove the decimal points. 35 - 31 is equal to 4, so that would make the difference 0.4.
Because both numbers are negative, the x must be a positive.
Therefor, 0.4 - 3.5 = -3.1
in february 2002 the argentine peso lost 70% of its value compared to the united states dollar. this devaluation drastically raised the price of imported products. according to a survey conducted by ac nielsen in april 2002, 68% of the consumers in argentina were buying fewer products than before the devaluation, 24% were buying the same number of products, and 8% were buying more products. furthermore, in a trend toward purchasing less-expensive brands, 88% indicated that they had changed the brands they purchased. f an appetizer, entree, beverage, and dessert. you have a choice of five appetizers, ten entrees, three beverages, and six desserts. how many possible complete dinners are possible?
Answers
After considering the given data we conclude that the there are 900 possible complete dinners generated that will be satisfactory to the dedicated question.
To evaluate the number of possible complete dinners, we need to apply multiplication regarding the number of choices for each course.
The number of possible complete dinners is the product of the number of choices for the appetizer, entree, beverage, and dessert.
Its is known to us that we have a choice of five appetizers, ten entrees, three beverages, and six desserts, the number of possible complete dinners is:
Number of possible complete dinners = number of choices for appetizer × number of choices for entree × number of choices for beverage × number of choices for dessert
Number of possible complete dinners = 5 × 10 × 3 × 6
Number of possible complete dinners = 900
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The complete question is
A restaurant menu has a price-fixed complete dinner that consists of an appetizer, an entrée, a beverage, and a dessert. You have a choice of 5 appetizers, 10 entrées, 3 beverages, and 6 desserts. Determine the total number of possible dinners.
Omar flies his plane out of a local airport. The airport is at an altitude of 750 feet above sea level. He flies his plane to an altitude of 1,600 feet above sea level. Which number represents sea level?]
Answers
Answer: zer0
Step-by-step explanation:
Answer:
0
Step-by-step explanation:
Find the component form of v given its magnitude and the angle it makes with the positive x-axis.
║v║ = 4, θ = 3.5°
Answers
To find the component form of v given its magnitude and the angle it makes with the positive x-axis, we can use the following formula , the component form of v is (3.9944, 0.2092) when its magnitude is 4 and it makes an angle of 3.5° with the positive x-axis.
We have ,
v = ║v║ (cos θ, sin θ)
where ║v║ is the magnitude of v, θ is the angle it makes with the positive x-axis, and (cos θ, sin θ) represents the direction of v in terms of the unit vector components along the x-axis and y-axis.
Substituting the given values, we get:
v = 4(cos 3.5°, sin 3.5°)
Using a calculator, we can find the cosine and sine values:
v = 4(0.9986, 0.0523)
Multiplying each component by 4, we get:
v = (3.9944, 0.2092)
Therefore, the component form of v is (3.9944, 0.2092) when its magnitude is 4 and it makes an angle of 3.5° with the positive x-axis.
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What solution to the system? Write the x-value and the y-value
10x+7y=1
-5x-7y=24
Answers
Answer:
y = - 7 and x = 5
Step-by-step explanation:
10x + 7y - 5x - 7y = 1 + 24
5x = 25
5x ÷ 5 = 25 ÷ 5
x = 5
- 5(5) - 7y = 24
- 25 - 7y = 24
- 25 + 25 - 7y = 24 + 25
- 7y = 49
- 7y ÷ - 7 = 49 ÷ - 7
y = - 7
please help! 2s+24 in factored form
Answers
Answer:
2(s + 12)
Step-by-step explanation:
Factor out the common factor 2: 2(s + 12)
Answer:
2(s+12)
Step-by-step explanation:
So what you want to find is a common factor. Then put it in the parentheses with the s.
Tawny wrote an expression to represent "the quotient of 82 and a number, decreased by 26." She then evaluated the expression when n = 2. Which statements are true about the expression and its value? Check all that apply.
Answers
Answer:
D, E, G
Step-by-step explanation:
Quotient of 82 and a number=82/n
Decreased by 26
82/n-26
She evaluate the expression when n=2
82/n-26
When n=2
82/2-26
=41-26
=15
Answer
D. The correct expression is 82/n-26
E.The operations include division and subtraction.
G.The value of the expression is 15
Answer:
The correct answer is b,d,e,g
Step-by-step explanation:
simplify each expression
Answers
Answer: Answer
Step-by-step explanation:
1. -83
2. 40
3. -38
4. 0
Please check my math!
Answer:
1 answer is39
2 answer is71
3 answer is38
4 answer is20/7
Step-by-step explanation:
Hope it is helpful for you
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Reed is currently r years old. Which expression represents his age seven years from now
Answers
Answer: r+7
Whatever his age is now, we add 7 to it. So we simply add 7 to r getting r+7.
Example: Say he is 10 years old now
r = 10
r+7 = 10+7 = 17
meaning he will be 17 seven years from now
AB is included between ______
Answers
AB is included between angle A and angle B.
Consider the differential equation (DE) ( + + ) + ( − ) =
3.1 Show that the DE is not exact.
3.2 Find an integrating factor for the DE.
3.3 Solve the DE.
Answers
3.1 Since ∂P/∂y ≠ ∂Q/∂x, the DE is not exact. 3.2 Integrating factor: μ(x) = e^(∫(-1 - 1) dx) = e^(-2x). 3.3 xy - 0.5e^(-2x) = C Where C is the constant of integration. This is the solution to the given differential equation (DE).
a. To show that the given differential equation (DE) is not exact, we need to check if the partial derivatives of its coefficients with respect to y and x are equal. Let's denote the DE as:
P(x, y) dx + Q(x, y) dy = 3
Now, we calculate the partial derivatives:
∂P/∂y = 1
∂Q/∂x = -1
Since ∂P/∂y ≠ ∂Q/∂x, the DE is not exact.
b. To find an integrating factor for the DE, we can use the formula: μ(x) = e^(∫(∂Q/∂x - ∂P/∂y) dx)
Plugging in the values, we have: μ(x) = e^(∫(-1 - 1) dx) = e^(-2x)
c. To solve the DE using the integrating factor, we multiply both sides of the DE by the integrating factor μ(x): e^(-2x) * [(∂P/∂y) dx + (∂Q/∂x) dy] = e^(-2x) * 3
This simplifies to: (e^(-2x) - y) dx + (-e^(-2x)) dy = 3 * e^(-2x) dx
To find the solution, we integrate both sides of the equation: ∫(e^(-2x) - y) dx + ∫(-e^(-2x)) dy = ∫(3 * e^(-2x)) dx
Integrating, we get: -0.5e^(-2x) - xy = -1.5e^(-2x) + C
Simplifying further, we have: xy - 0.5e^(-2x) = C
Where C is the constant of integration. This is the solution to the given differential equation (DE).
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56 = ?×?×?×? i want to know what number ecual yo 56 by multiplying it 4 times to get your answer
Answers
Answer:
4 * 14 = 56
Step-by-step explanation:
56/2 = 28
28/2 = 14
14/2 = 7
7/7 = 1
56 = 2 * 2 * 2 * 7
56 = (2 * 2) * (2 * 7)
56 = 4 * 14
Which of the following would be considered full-time work? forty hours forty hours fifty-one hours fifty-one hours thirty-three hours thirty-three hours forty-five hours
45 hours, 40 hours, 51 hours
Answers
Out of the options listed, both 40 hours and 45 hours would be considered full-time work.
Determining hours of full-time workWhat can be considered as full-time work vary from country to county and also from industry to industry. Generally, full-time work is usually defined as working a certain number of hours per week, typically between 35 and 40 hours.
Therefore, out of the options given, both 40 hours and 45 hours would be considered full-time work. 51 hours is generally considered to be more than full-time work, and it may be considered overtime in many industries.
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(−2) × 36 × (−5) = ______.
Answers
the answer is 360 Hope it helps you
What comes after a pilot project?.
Answers
According to question, project managers can assess the outcomes to identify any hazards the project might encounter after it is completely executed.
These risks are frequently examined by project managers and stakeholders to decide whether they are controllable and whether it is worthwhile to move on with the entire project. An organization can control the risk of a novel concept and spot any flaws through a pilot project before significant resources are spent.
On the other hand, a trial is a limited implementation before you launch the pilot project broadly. All the advantages of a pilot project are present, but they are more thorough, and the team is frequently able to test out more features and observe the results on a wider scale.
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Would you like a corndog?
Answers
Answer:
a corndog? mkk why not I guess
Answer:
No
Step-by-step explanation:
What is -7+3z+4z simplified?
Answers
Answer:
The answer is 7z-7. Hope this helped!
Step-by-step explanation:
please help thank you
Answers
Answer:
B.
Step-by-step explanation:
b. because its the L in the cearl bowl
Answer:
g (grams)
Step-by-step explanation:
Example
Jesse needs to decide which metric unit he should use to weigh a bowl of books and documents.
First, think about the size and weight of the objects in the bowl.
Milligrams would be far too small of a unit. The books alone would probably weigh a few million milligrams. Even grams is really not ideal, since the total weight would be thousands of grams.
What is a response variable in stats?
Answers
A response variable in statistics is a variable that is being measured or observed in a study. It is thought to be affected by the explanatory variable.
Response variable in statistics is a variable that is being measured or observed in a study. It is also known as the dependent variable or the outcome variable.
This variable is thought to be affected by the explanatory variable, also known as the independent variable.
This allows us to understand if the changes involved are significantly related or not or whether it is due to certain factors.
For example, if a study is examining the effect of exercise on weight loss, the response variable would be weight loss and the explanatory variable would be exercise. The response variable is typically plotted on the y-axis of a graph, while the explanatory variable is plotted on the x-axis.
In summary, a response variable in statistics is a variable that is being measured or observed in a study.
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HELPPP!!! BRIANLEST???
Answers
Answer:
5/18
Step-by-step explanation:
Irresponsible and 0 siblings (5) over total (18)
5/18 doesn't simplify
A bacteria culture starts with 160 bacteria and grows at a rate proportional to its size. After 5 hours
there will be 800 bacteria.
Answers
The population after t hours is 160 x (1.3797)^t.
What is the population after t hours?The first step is to determine the growth rate of the bacteria:
growth rate = (future population / initial population) ^(1/number of years) - 1
(800 / 160)^(1/5) - 1= 37.97
The exponential function : 160 x (1.3797)^t
Here is the complete question:
A bacteria culture starts with 160 bacteria and grows at a rate proportional to its size. After 5 hours
there will be 800 bacteria. Express the population after t hours as a function of t. population:
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.What kind of slope does perpendicular lines have?
Answers
Answer:
negative reciprocals
Step-by-step explanation:
On a scale drawing a building measures 4.5 cm tall is the scale of the drawing is 25 m per2 cm how tall is the actual building
Answers
Answer:
Actual height of the building \(=112.5\) meters
Step-by-step explanation:
Given
Scale of the drawing
1 cm = 25 meters
This means that any 1 cm line on the map represents a length of 25 meters in actual.
The height of the building on map \(= 4.5\) cm
Actual height of the building
\(= 4.5 * 25\\=112.5\)meters
in a math class, tests are 50% of your grade, homework is 30% of your grade, test reviews are 5% of your grade, and the final exam is 15% of your grade. at the end of the semester, your test average is 59, your homework average is 90, and your review average is 72. what score do you need on the final exam to get a c (72%) in the class? (round your answer to 2 decimal places.)
Answers
The score needed on the final exam to achieve a C grade is 79.33, rounded to 2 decimal places.
Homework is worth 30% of the grade, which is also a significant portion. The average homework score is 90, which means the average of all homework scores is 90.
Test reviews are only worth 5% of the grade, so they have the smallest impact on the overall average. The average test review score is 72, which means the average of all test review scores is 72.
Finally, the final exam is worth 15% of the grade, and we need to find out what score is needed on the final exam to achieve a C grade, which is 72%.
To calculate the overall average, we can use the weighted average formula.
Overall Average = (Tests Average x 0.5) + (Homework Average x 0.3) + (Test Reviews Average x 0.05) + (Final Exam Score x 0.15)
Substituting the values we know, we get:
Overall Average = (59 x 0.5) + (90 x 0.3) + (72 x 0.05) + (Final Exam Score x 0.15)
Simplifying this equation, we get:
Overall Average = 29.5 + 27 + 3.6 + (Final Exam Score x 0.15)
Overall Average = 60.1 + (Final Exam Score x 0.15)
To get a C grade, the overall average needs to be 72%. So we can set up an equation and solve for the Final Exam Score:
72 = 60.1 + (Final Exam Score x 0.15)
Solving for Final Exam Score, we get:
Final Exam Score = (72 - 60.1) / 0.15
Final Exam Score = 79.33
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Find the equation of this line
Answers
Answer:
y=4x-9
Step-by-step explanation:
-9=y intercept
up 4 over 1
4/1=4
slope =4
Calculate the slope of the line that contains:
(4, 5) and (-4, 3)
Answers
The reactions of the pyruvate dehydrogenase complex are required to generate the substrate that is fed into the TCA (Kreb's) cycle from pyruvate. The 3 enzymes that make up this complex are pyruvate dehydrogenase (E1), dihydrolipoyl transacetylase (E2) dihydrolipoyl dehydrogenase (E3). a. Name the one diffusible reaction product (i.e. the product that is free to leave the enzyme complex) of each enzyme of the complex. b. Draw the "business end" of the fully reduced form of lipoic acid. c. Using words, fully describe the function of E3 in this complex. Your answer should include all cofactors used, all intermediates and products of this enzyme. DO NOT show any mechanisms for this part.
Answers
The product that can leave the enzyme complex for each enzyme in the complex are: CoA for Pyruvate dehydrogenase (E1), Acetyl group for Dihydrolipoyl transacetylase (E2), and NADH for Dihydrolipoyl dehydrogenase (E3).
The "business end" of the fully reduced form of lipoic acid is shown in an illustration. The function of E3 in the complex is to oxidize dihydrolipoamide with NAD⁺, contributing to the process of oxidative phosphorylation.
a. The product that is free to leave the enzyme complex of each enzyme in the complex are:
Pyruvate dehydrogenase (E1): CoA, which is free to leave the enzyme complex after the pyruvate has been oxidized.
Dihydrolipoyl transacetylase (E2): Acetyl group, which is free to leave the enzyme complex after it has been transferred to CoA.
Dihydrolipoyl dehydrogenase (E3): NADH, which is free to leave the enzyme complex after dihydrolipoamide has been oxidized.
b. The "business end" of the fully reduced form of lipoic acid can be drawn as shown below:
Illustration
c. The function of E3 in this complex is to oxidize the dihydrolipoamide with NAD⁺. The reduced dihydrolipoamide is reoxidized by E3 in the following reaction:
Dihydrolipoamide + FAD + NAD⁺ → Lipoamide + FADH₂ + NADH + H⁺
Where FAD is the cofactor that E3 utilizes. FADH₂ is later oxidized by ubiquinone in the electron transport chain. Therefore, E3 contributes to the process of oxidative phosphorylation.
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Help help /ayuda ayudaaa༎ຶ‿༎ຶ!!!!!!!!!!!!
find the length of the missing side of the triange shown below
round to the nearest tenth if necessary
Answers
Answer:
Step-by-step explanation:
Use Pythagorean theorem
Base² + altitude² = hypotenuse²
Base² + 7² = 20²
Base² + 49 = 400
Base² = 400 - 49 = 351
Base = √351 = 18.73 = 18.7 mm