LetY1,Y2,Yn denote a random sample of size n from a population whose density is given by f(y)={αyα−1θα,0≤y≤θ0,elsewhere,where α>0 is a known, fixed value, but θ is unknown. Consider the estimator ˆθ=max(Y1,Y2,...Yn).(a) Show that ˆθ is a biased estimator for θ.(b) Find a multiple of ˆθ that is an unbiased estimator of θ.(c) Derive MSE(ˆθ).
Answers
(a) θ is a biased estimator for θ.
(b) (α+1)Y/α is an unbiased estimator of θ.
(c) MSE(θ) = αθ^2/[(α+1)^2(α+2)]
(a) To show that θ is a biased estimator for θ, we need to show that E(θ) ≠ θ.
Using the definition of the maximum function, we have
P(θ ≤ y) = P(Y1 ≤ y, Y2 ≤ y, ..., Yn ≤ y) = (F(y))^n
where F(y) is the cumulative distribution function of Y.
Differentiating both sides with respect to y, we get:
f(θ) = n(F(θ))^(n-1)f(θ)
Simplifying, we get
F(θ) = (1/n)^(1/(n-1))
Using this result, we can find the expected value of θ
E(θ) = ∫₀^∞ θf(θ)dθ = ∫₀^θ θαθ^α-1dθ = αθ/(α+1)
Thus, E(θ) ≠ θ, which means that θ is a biased estimator for θ.
(b) To find a multiple of θ that is an unbiased estimator of θ, we can use the method of moments.
We know that the population mean of Y is
μ = ∫₀^θ yf(y)dy = αθ/(α+1)
The sample mean is
Y = (Y1+Y2+...+Yn)/n
Equating these two expressions and solving for θ, we get
θ = (α+1)Y/α
Thus, (α+1)Y/α is an unbiased estimator of θ.
(c) The mean squared error (MSE) of θ can be written as
MSE(θ) = E[(θ - θ)^2]
Expanding the square and using the linearity of expectation, we have
MSE(θ) = E[θ^2] - 2θE[θ] + E[θ]^2
We already know that E[θ] = αθ/(α+1).
To find E[θ^2], we can use the fact that θ = max(Y1,Y2,...Yn)
P(θ ≤ y) = P(Y1 ≤ y, Y2 ≤ y, ..., Yn ≤ y) = (F(y))^n
Differentiating both sides with respect to y, we get
f(θ) = n(F(θ))^(n-1)f(θ)
Using this result, we can find E[θ^2]
E[θ^2] = ∫₀^∞ θ^2f(θ)dθ = ∫₀^θ θ^2αθ^α-1dθ = αθ^2/(α+2)
Substituting these expressions into the MSE formula, we get
MSE(θ) = αθ^2/(α+2) - 2θ(αθ/(α+1)) + (αθ/(α+1))^2
Simplifying, we get
MSE(θ) = αθ^2/[(α+1)^2(α+2)]
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Related Questions
A marine biologist measured one dish that was 1 1/4 of a foot and a second fish that was 3/4 of a foot long. How much longer was the first fish
Answers
Answer:
1/3 foot longer
Step-by-step explanation:
2/3-1/3=1/3
QUESTION 1 Determine the general solution of: 2 sin x. cos x = COS X
Answers
Starting with the given equation: 2 sin x cos x = cos x
By dividing both sides by cos x, we may simplify: 2 sin x = 1
Dividing both sides by 2:
sin x = 1/2
The values of x that fulfil sin x = 1/2 must now be found, and they are:
x = π/6 + 2πn or x = 5π/6 + 2πn, where n is any integer.
Therefore, the general solution is:
x = π/6 + 2πn or
x = 5π/6 + 2πn, where n is any integer.
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BRAINLIEST!! What’s 36.3 square root ?
Answers
Answer:
6.02
Step-by-step explanation:
What is 8c + c = , is it 8cc, 9c, 8c2, 8c
Answers
Answer:
9c
Step-by-step explanation:
if you take 8c and add another c, it equals 9c
Answer:
9C
Step-by-step explanation:
Find x in the given figure.
Answers
So let’s calculate “6”: 180° - 55° = 125°
Therefore x = 125°
margo borrows $900.00 and must pay it back with 6% interest after 7 months. what is the total amount margo will pay back ?
Answers
Answer:
$954
Step-by-step explanation:
6% = 0.06
Adding 1 to the interest makes it so it’s added to the original amount.
$900 x 1.06 = $954
What is the answer to the question??
Answers
Answer:you need to show the diagram in order for me to
Step-by-step explanation:
The factory wants you to build a box that will hold twice as many cubes. What are the dimensions of a box that contains two times as many cubes as a box that is 2 by 3 by 5? Write the dimensions and explain how you found the answer.
Answers
A box that contains two times as many cubes as a box that is 2 by 3 by 5 can have dimensions of 2 by 3 by 10.
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
A box that is 2 by 3 by 5 has a total of 2 x 3 x 5 = 30 cubes. To build a box that will hold twice as many cubes, we need to find the dimensions that will give a total of 60 cubes.
Let's assume the dimensions of the new box are x, y, and z. Then, we have:
xyz = 60
We can express one of the variables in terms of the other two.
z = 60 / (xy)
Substitute this in above equation.
xy60/xy=60
60=60
This is true for any values of x and y, as long as z is calculated as z = 60 / (x y).
There are infinitely many solutions for the dimensions that will give a box with a total of 60 cubes.
set of dimensions that satisfies the condition is x = 2, y = 3, and z = 10.
Hence, a box that contains two times as many cubes as a box that is 2 by 3 by 5 can have dimensions of 2 by 3 by 10.
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(URGENT)An engineer has a 40:1 scale drawing of a bridge. The dimensions of the scaled bridge deck are 24 inches by four and four fifths inches. What is the area of the actual bridge deck in square feet?
460 square feet
640 square feet
960 square feet
1,280 square feet
Answers
The area οf the actual bridge deck in square feet is 1,280 square feet.
What is Area?Area refers tο the measurement οf the size οf a twο-dimensiοnal surface, usually measured in square units such as square inches, square feet, οr square meters. It represents the amοunt οf space that is inside the bοundary οf a flat οbject οr shape.
The scale οf the drawing is 40:1, which means that the dimensiοns οf the scaled bridge deck are 40 times smaller than the actual bridge deck. Tο find the area οf the actual bridge deck in square feet, we need tο cοnvert the dimensiοns οf the scaled bridge deck frοm inches tο feet and then multiply by the square οf the scale factοr (40² = 1600).
The dimensiοns οf the scaled bridge deck in feet are:
24 inches = 2 feet
4 and 4/5 inches = 0.4 feet
Sο, the area οf the scaled bridge deck in square feet is:
2 feet x 0.4 feet = 0.8 square feet
Tο find the area οf the actual bridge deck in square feet, we need tο multiply the area οf the scaled bridge deck by the square οf the scale factοr:
Area οf actual bridge deck = 0.8 square feet x (40²) = 1280 square feet
Therefοre, the area οf the actual bridge deck in square feet is 1,280 square feet.
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Answer:
Step-by-step explanation:
The scale factor indicates the actual bridge is 40 times larger than the drawing.
Increase the scale dimensions by the factor of 40.
24 times 40 = 960"
4.8 times 40 = 192" (I changed 4 4/5 to 4.8)
Area = length times width
960 times 192 = 184320 sq inches actual bridge deck in square inches
Divide 183320 by 144 (sq inches in a square foot) = 1280 sq feet
The Gordon family plans to buy a TV. One TV has a purchase price of $330 and an estimated yearly operating cost of $14. The other has a purchase price of $369 and an estimated yearly operating cost of $9. Which TV should the Gordons buy if they plan to keep it for 8 years? Use rational functions to help justify your answer
Answers
Answer:
Step-by-step explanation:
They should purchase the $369 one with the $9 plan.
330 + (14x8) =442
369+ (9x8) = 441
Hope this helps!
Answer:
rational functions that model the average annual cost of each television if it is owned for x years, such as f(x)= (330+14x)/x and
g(x)= (369+9x)/x
calculation proving that over 8 years, the annual cost for the $330 television is $55.25; for the $369 television, it is $55.13
statement that the Gordons should buy the $369 television because it is less expensive overall
Step-by-step explanation:
edge 2022
Bishwant thought of a number
He doubled it and added 5.if
he got 13, find the number
thought.
Answers
Answer:
The number Bishwant thought of is 4.
Step-by-step explanation:
First, let's think about what we can eliminate.
If Bishwant doubled the number and added 5, getting 13--then the number he thought of is obviously less than 13, 12, 11, 10, 9, 8, and 7.
Now, let's do some trials!
6 + 6 = 12; 12 + 5 = 17 <-- Incorrect
5 + 5 = 10; 10 + 5 = 15 <-- Incorrect
4 + 4 = 8; 8 + 5 = 13 <--Ah!
4 is the answer! (If you have any questions about this riddle, feel free to comment--but, please stay respectful everywhere, all the time.)
Find the value of K when K = 3n2 + w when n = 2 and w = - 10
Answers
Answer:
K = 2
Step-by-step explanation:
substitute n = 2 and w = - 10 into the equation.
K = 3n² + w = 3(2)² + (- 10) = 3(4) - 10 = 12 - 10 = 2
Please help I’m stuck and keep getting the wrong answer
Answers
The time spent higher than 26 meters above the ground is 0.42 minutes. Answer: 0.42
A Ferris wheel is 30 meters in diameter and boarded from a platform that is 4 meters above the ground.
The six o'clock position on the Ferris wheel is level with the loading platform.
The wheel completes 1 full revolution in 2 minutes.
We have to find how many minutes of the ride are spent higher than 26 meters above the ground.
So, let's start with some given data,Consider the height of a person at the six o'clock position = 4 meters
So, the height of a person at the highest point = 4 + 15 = 19 meters (since the diameter is 30 meters, the radius will be 15 meters)
Also, the height of a person at the lowest point = 4 - 15 = -11 meters
Therefore, the Ferris wheel completes one cycle from the lowest point to the highest point and back to the lowest point.
So, the total distance travelled will be = 19 + 11 = 30 meters.
Also, we are given that the wheel completes 1 full revolution in 2 minutes.
We need to calculate the time spent higher than 26 meters above the ground.
So, the angle between the 6 o'clock position and 2 o'clock position will be equal to the angle between the 6 o'clock position and the highest point.
This angle can be calculated as follows:
Angle = Distance travelled by the Ferris wheel / Circumference of the Ferris wheel * 360 degrees
Angle = 30 / (pi * 30) * 360 degrees
Angle = 360 degrees / pi
= 114.59 degrees
So, the total angle between the 6 o'clock position and the highest point is 114.59 degrees.
Now, we need to find out how much time is spent at an angle greater than 114.59 degrees.
This can be calculated as follows:
Time = (Angle greater than 114.59 degrees / Total angle of the Ferris wheel) * Total time taken
Time = (180 - 114.59) / 360 * 2 minutes
Time = 0.42 minutes
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Pls help Darnell is an election officer. On election day, he travels to the polling place, which is 4.4 miles away from his home. On a map of Harrison County, these two places are 8 inches apart. What is the scale of the map? Write your answer in simplest form using whole numbers.
Answers
Answer:
The scale of the map is 1 : 34848---------------------------------
Real distance is 4.4 miles and on the map it is 8 inches.
The scale is:
8 in : 4.4 miles = Divide both sides by 81 in : 0.55 milesor
1 in : 0.55 * 63360 in = Convert 1 mile = 63360 inches1 : 34848Solve this system by elimination
10x+6y=14
-x-6y=-23
Show step by step
Answers
Answer:
(-1, 4)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDASEquality PropertiesAlgebra I
Solving systems of equations using substitution/eliminationStep-by-step explanation:
Step 1: Define systems
10x + 6y = 14
-x - 6y = -23
Step 2: Solve for x
Elimination
Add 2 equations together: 9x = -9Divide 9 on both sides: x = -1Step 3: Solve for y
Define original equation: -x - 6y = -23Substitute in x: -(-1) - 6y = -23Multiply: 1 - 6y = -23Subtract 1 on both sides: -6y = -24Divide -6 on both sides: y = 44x +y = 8
2x - 3y = 4
There is no solution.
® There are infinitely many solutions.
© There is exactly one solution at (2,0).
There is exactly one solution at (0,2).
Pick a statement that describes the solution correctly
Answers
Hello! Your answer will be the second provided answer.
See attachment below for explanation...
Find the value of x.
Answers
Answer:
5
Step-by-step explanation:
explain the types of frequency distribution in statistics
Answers
The two types of frequency distributions are Discrete Frequency Distribution and Grouped Frequency Distribution
What are the types of distribution?The two types of frequency distributions are;
Discrete Frequency Distribution:Grouped Frequency DistributionWhen the data consists of discrete, separate values, this sort of distribution is used. It displays the frequency or number of occurrences of each value.
The data is often expressed in a table with two columns: one for distinct values and another for the frequencies of those values.
This distribution is utilized when the data is continuous and has a wide range of values. It entails categorizing the data into intervals or classes and then calculating the frequency of values that fall within each interval.
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-4n>5 match inequality to it corresponding statement
Answers
Answer:
n<-1.25
Step-by-step explanation:
A group of people were asked if they had run a red light in the last year. 186 responded "yes" and 130 responded "no." Find the probability that if a person is chosen at random from this group, they have not run a red light in the last year.
Answers
Step-by-step explanation:
There are 186 + 130 = 316 people total, Out of those 316, there are 186 people who responded "yes" to running a red light.
so the probablity is approximately 186/316 = 0.58860759493
If you want this probablity as percentage , then it would be approximately
58.860759493%
Thank You
Help meeeeeeeeeeeeeeee
Answers
Answer:
75
Step-by-step explanation:
f(4) means to use 4 in place of x.
f(x) = 3(5)^(x-2)
fill in 4 for
f(4) = 3(5)^(4-2)
do the 4-2 subtraction first
f(4) = 3(5)^2
do the 5 to the 2nd power next.
f(4) = 3(25)
last, multiply.
f(4) = 75
This is just following the typical order of operations.
tim wants to order pizza for 22 employees.Each employee should get 1/4 of a pizza.How many pizzas should tim order ?
Answers
Tim should order approximately 6 pizza.
Define division.Division in mathematics is the process of dividing an amount into equal parts. For instance, we may split a group of 20 people into four groups of 5, five groups of 4, and so on. One of the four fundamental arithmetic operations, or how numbers are combined to create new numbers, is division. The other operations are multiplication, addition, and subtraction. Mathematicians use addition, subtraction, multiplication, and division as their four fundamental arithmetic operations. The division is one of these four operations that we employ most frequently in our daily work. It involves dividing a huge group into equally sized smaller units. Divide 25, for instance, by 5.
Given Data
Number of employees = 22
Slice of pizza one should get = 1/4
Dividing 22 by 1/4
\(\frac{22}{4}\)
5 and \(\frac{1}{2}\)
Tim should order approximately 6 pizza.
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What is twenty thousand, six hundred two in expanded form?
Simplify (2+3)^2-16divided by 2
Answers
Answer:
20,000+600+2
4 1/2
Step-by-step explanation:
(2+3)^2-16 divided by 2
first you 2+3=5 then 5 times 5 equals 25
Then 25-16=9
9/2=4 1/2
A tile factory earns money by charging a flat fee for delivery and a sales price of $0.25 per tile. One customer paid a total of $3,000 for 10,000 tiles. The equation y – 3,000 = 0.25(x – 10,000) models the revenue of the tile factory, where x is the number of tiles and y is the total cost to the customer.
Which function describes the revenue of the tile factory in terms of tiles sold?
Answers
The answer is 1.C 2.$500
x-2y+z=2
bx+ay=3
-2y+z=-4
Choose appropriate values for a and b
Answers
Answer:
Attached
Step-by-step explanation:
Just for your reference, this question will get unlimited number of solutions, attached is a few examples I calculated from Excel. Have fun to pick one or create one to be your favorite.
Your investment adviser tells you that Acme Widgets has a 40% chance of increasing your investment by $10,000, and Gimbles Stuff has a 30% chance of increasing your investment by $15,000. Which has the higher Expected Value?
Answers
Explanation: 15,000x0.3= 4,500 whereas 10,000x0.4= 4,000
Please help! Thank you, Picture down Below!
Answers
Answer:
A.24
Step-by-step explanation:
that is the.value of x
Answer:
(5x + 15)⁰+ (3x - 3⁰) = 180⁰ ( same side on interior angle theorm)
(8x + 12)⁰= 180⁰
8x = 168⁰
x = 21
Hope it helps u
Simplify the following expression: a + 2c − 5b − c + a − b
Answers
Answer:
Step-by-step explanation:
an 18, 2016 - A + 2c − 5b − c + a − b1a + 1a + 2c − 1c − 5b − 1b(1+1) a + (2-1)c + (−5−1)b 2a + 1c − 6bAnswer: 2a + c − 6b.
2 answers
Each of 6 students reported the number of movies they saw in the past year. Here is what they reported. 14, 11, 15, 19, 12, 10 Send data to calculator Find the mean number of movies that the students saw. If necessary, round your answer to the nearest tenth. movies X Ś
Answers
The mean number of movies that the students saw is 13.5 (rounded to the nearest tenth).
To find the mean number of movies the students saw, you need to calculate the average of the given data. Here are the reported numbers of movies seen by the 6 students: 14, 11, 15, 19, 12, 10.
To calculate the mean, you sum up all the reported numbers and divide by the total number of students. In this case, the total number of students is 6.
So, let's calculate the mean:
(14 + 11 + 15 + 19 + 12 + 10) / 6 = 81 / 6 = 13.5
Therefore, the mean number of movies that the students saw is 13.5 (rounded to the nearest tenth).
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What is the equation of a circle with center (1,-4) and radius 2?
Answers
Step-by-step explanation:
Centre of circle(h,k)=(1,-4)
radius of the circle(r)=2
Equation of the circle is,
(x-h)^2+(y-k)^2=r^2
(x-1)^2+(y+4)^2=2^2
x^2-2x+1+y^2+8y+16=4
x^2+y^2-2x+8y+13=0 is the req.equation of the circle.
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