Let Y 1 ,Y2 and Y3 be independent and identically distributed random variables with expectation E[Yi ]=μ and variance V[Yi ]=σ 2
for i=1,2,3. Suppose we want to estimate the expectation μ, and we propose to combine our three observations Y 1 ,Y2 and Y3 by defining a weighted average: Yˉω =ω1 Y1+ω 2Y2 +ω3 Y3 where ω1 ,ω2 and ω3
are constants between 0 and 1 . 1 (a) Calculate E[ Y
ˉ
ω
]. What condition do the weights ω 1
,ω 2
and ω 3
need to satisfy for Y
ˉ
ω
to be an unbiased estimator for μ ? (b) Calculate V[ Y
ˉ
ω
]. Using the condition you found on (a), find the set of weights that minimize the variance of Y
ˉ
ω
. Justify your answer. Now suppose that observations Y 1
,Y 2
and Y 3
are independent and have the same mean μ, but they have different variances. Specifically, V[Y 1
]=V[Y 2
]=σ 2
and V[Y 3
]=2σ 2
Answers
a.) The weights, ω1 + ω2 + ω3 = 1. b.) The weights, V[Y_bar ω] = ω1²V[Y1] + ω2²V[Y2] + ω3²V[Y3] = ω1²σ² + ω2²σ² + ω3²σ² = σ²(ω1² + ω2² + ω3²). To estimate the expectation μ, we propose combining three independent and identically distributed random variables Y1, Y2, and Y3 using a weighted average, Y_bar ω = ω1Y1 + ω2Y2 + ω3Y3.
(a) The weights ω1, ω2, and ω3 need to satisfy the condition ω1 + ω2 + ω3 = 1 for Y_bar ω to be an unbiased estimator for μ. The expectation E[Y_bar ω] is calculated as E[Y_bar ω] = ω1E[Y1] + ω2E[Y2] + ω3E[Y3] = ω1μ + ω2μ + ω3μ = (ω1 + ω2 + ω3)μ. For Y_bar ω to be an unbiased estimator for μ, its expectation should equal μ. Therefore, ω1 + ω2 + ω3 = 1.
(b) The variance V[Y_bar ω] can be calculated as V[Y_bar ω] = V[ω1Y1 + ω2Y2 + ω3Y3]. Since Y1, Y2, and Y3 are independent, the variance of their sum is the sum of their variances. Therefore, V[Y_bar ω] = ω1²V[Y1] + ω2²V[Y2] + ω3²V[Y3] = ω1²σ² + ω2²σ² + ω3²σ² = σ²(ω1² + ω2² + ω3²). The variance of Y_bar(ω) can be calculated, and by applying the condition from (a), the set of weights that minimize the variance can be determined.
To minimize the variance, we can apply the condition from (a), ω1 + ω2 + ω3 = 1, and use a technique called Lagrange multipliers. By introducing the Lagrange multiplier λ, we form the Lagrangian L = σ²(ω1² + ω2² + ω3²) + λ(ω1 + ω2 + ω3 - 1). Taking partial derivatives with respect to ω1, ω2, ω3, and λ, we can solve the equations to find the set of weights that minimize the variance. The specific values will depend on the given values of σ² and the constraints imposed by the problem.
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Related Questions
6) Which list shows how many equal-sized groups of 18 students Ms. Lee can make?
1, 2, 3, 4, or 6
7, 9, 14, 18, or 28
1, 2, 3, 6, 9, or 18
1, 2, 4, 7, 14, or 28
Answers
Answer:
1,2,3,6,9
Step-by-step explanation:
Got it right
Solve for x in the triangle. Round your answer to the nearest tenth.
6
X
69°
X
Answers
hello i am just need points so sorry you didnt get answer
Q 42 - The proportion of salary of An and B is 5:3 and that of their use is 9:5. On the off chance that they spare Rs. 2600 and Rs. 1800, then their livelihoods are: A-9000, 5400 B-10000, 6000 C-6000, 3600
Answers
Answer: Let's solve this problem step by step.
First, let's assume the salaries of An and B to be 5x and 3x, respectively, where x is a common multiplier.
According to the given information, they save Rs. 2600 and Rs. 1800, respectively. Since savings come from the remaining portion of their incomes after spending, we can calculate their expenditures as follows:
For An:
Income of An = Salary of An + Savings of An
Income of An = 5x + 2600
For B:
Income of B = Salary of B + Savings of B
Income of B = 3x + 1800
Now, let's consider the proportion of their expenditures. It is given that the proportion of their expenditures is 9:5. So, we can write the following equation:
(Expenditure of An)/(Expenditure of B) = 9/5
Since expenditure is the complement of savings, we have:
[(Income of An - Savings of An)] / [(Income of B - Savings of B)] = 9/5
Substituting the previously derived expressions for income, we get:
[(5x + 2600 - 2600)] / [(3x + 1800 - 1800)] = 9/5
Simplifying the equation, we have:
5x / 3x = 9/5
Cross-multiplying, we get:
5 * 3x = 9 * 3x
15x = 27x
Subtracting 27x from both sides, we have:
0 = 12x
This implies that x = 0, which is not a valid solution. Therefore, there seems to be an error or inconsistency in the given information or equations. Please recheck the problem statement or provide additional information to help resolve the issue.
What is the area of δabc? round to the nearest tenth of a square unit. 3.9 square units 8.4 square units 11.8 square units 17.7 square units
Answers
The triangle ABC has a surface area of 17.7 square units. Rounding to the nearest tenth of a square unit,
Heron's Formula can be used to calculate the triangle ABC's area. The equation reads as follows:
A = √[s(s-a)(s-b)(s-c)]
Where s is one-half of the triangle's circumference and a, b, and c are its three sides.
By summing the triangle's sides, we can determine the perimeter:
P = a + b + c
The formula below is used to get the other half of the perimeter.
s = P/2
For the triangle ABC, we have the sides a=3, b=5, and c=4.
Therefore, half of the perimeter is s = 8.
Now, Heron's formula can be used to determine the triangle ABC's surface area:
A = √[s(s-a)(s-b)(s-c)]
= √[8(8-3)(8-5)(8-4)]
= √[8(5)(3)(4)]
= 17.7 square units
Therefore, the triangle ABC has a surface area of 17.7 square units, Rounding to the nearest tenth of a square unit.
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simplify fully the following :
2√3 x 3√8
Answers
Answer:
29.3938769134
Step-by-step explanation:
Answer:
\(\sf 12\sqrt{6}\)
A 150 -day \( \$ 4 \) million CD has a \( 3.95 \) percent annual rate quote. If you buy the CD. how much will you collect in 150 days? \( \$ 4,047,439 \) \( \$ 4,045.678 \) \( \$ 4,065,833 \) \( \$ 4,
Answers
If you buy the $4 million CD with a 3.95% annual interest rate and hold it for 150 days, you will collect $4,047,439 at maturity.
To calculate the amount you will collect after 150 days, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = time (in years)
In this case, the principal amount is $4 million, the annual interest rate is 3.95% (or 0.0395 as a decimal), and the CD has a 150-day term.
To determine the number of compounding periods in 150 days, we need to consider the compounding frequency. CD interest is typically compounded daily. Therefore, we have:
n = 365 (number of days in a year) / 150 (number of days in the CD term) = 2.4333
Plugging the values into the formula:
A = $4,000,000 * (1 + 0.0395/2.4333)^(2.4333*150/365)
A = $4,047,439
If you buy the $4 million CD with a 3.95% annual interest rate and hold it for 150 days, you will collect $4,047,439 at maturity.
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V (x) = x(9-2x) (11-2x)
Answers
The expanded form of V(x) = x(9-2x) (11-2x) is V(x) = 4x^3 - 20x^2 + 99x
How to expand the expression?The missing information in the question is to express V(x) in standard form
We have:
V(x) = x(9-2x) (11-2x)
Open the brackets
V (x) = x(99 - 2x - 18x + 4x^2)
Evaluate the like terms
V (x) = x(99 - 20x + 4x^2)
Expand
V (x) = 99x - 20x^2 + 4x^3
Rewrite as:
V(x) = 4x^3 - 20x^2 + 99x
Hence, the expanded form of V(x) = x(9-2x) (11-2x) is V(x) = 4x^3 - 20x^2 + 99x
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prove that there are no solutions in integers x and y to the equation 2x2 + 5y2 = 14.
Answers
The quadratic equation does not have solutions where both x and y are integers.
How to prove that there are no integer solutions for the equation?Here we have the following quadratic equation:
2x^2 + 5y^2 = 14
We want to check that we can't solve this equation with x and y integers, to see that, we can start by isolating one of the variables:
2x^2 + 5y^2 = 14
5y^2 = 14 - 2x^2
y^2 = (14 - 2x^2)/5
y = ±√((14 - 2x^2)/5)
Now, trivially:
(14 - 2x^2)/5
Is not an integer for any integer value of x, and this happens because 5 is not a factor of 14, thus, for any integer value of x the value of y that we get is non integer, then there aren't two integers that solve the quadratic equation.
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Type the correct answer in each box. Use numerals instead of words.
Find the missing side and angle measures in triangle ABC. Round your answers to the nearest tenth.
A
40°
B
23
The measure of angle B is approximately
The measure of angle C is approximately
The length of side AB is approximately
15
units
Answers
From the calculation that we have done;
Angle B = 80.2°
Angle C = 59.8°
Side c = 20.3
What is the sine rule?The sine rule, often referred to as the law of sines, is a geometrical concept that links the sines of a triangle's opposite angles to the lengths of its sides. It claims that for all triangle sides and angles, the ratio of a side's length to the sine of its opposite angle is constant.
We have to apply the sine rule in the calculation;
We know that;
a/Sin A = b/SinB
Hence;
15/Sin 40 = 23/SinB
15 Sin B = 23Sin 40
B = Sin-1( 23Sin 40/15)
B = 80.2°
Angle C =
180 - (80.2 + 40)
= 59.8°
Then Side c
c/Sin 59.8 = 15/Sin 40
cSin40 = 15 Sin 59.8
c = 15 Sin 59.8/Sin 40
= 12.96/0.64
= 20.3
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When creating a right triangle using 3 squares as seen in the online video, which sides create the right angle in the triangle?
Answers
When creating a right triangle using three squares, the sides of the two squares with the smallest areas will create the right angle in the triangle (see image attached below).
Creating a Right Triangle with Three SquaresThe right angle of a right triangle is formed by the two small legs of the right triangle that are perpendicular to each other.
Just like in the diagram attached below, the two squares with the smallest sides will form the right angle in the right triangle.
Therefore, we can conclude that when creating a right triangle using three squares, the sides of the two squares with the smallest areas will create the right angle in the triangle.
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Please someone help me
Answers
Answer:
D
Step-by-step explanation:
Answer:
uhh what kinda symbols r those
Step-by-step explanation:
if a random variable x has a mean 1 and a variance 4, then the average of the 50 samples of x has mean of a) 1/50 b) 2/50 c) 4/50 d) 1
Answers
Option (a) 1/50 The average of the 50 samples of x can be calculated as the sum of the values of x divided by the number of samples, which is 50 in this case.
Since x has a mean of 1, we know that the sum of the values of x is 50 times 1, which is 50. The variance of x is 4, which means that the standard deviation of x is 2. This means that the standard error of the mean of x, which is the standard deviation of the sample mean, is 2 divided by the square root of 50, which is approximately 0.283. Therefore, the 95% confidence interval for the mean of the 50 samples of x is [1 - 1.96(0.283), 1 + 1.96(0.283)] = [0.44, 1.56]. Since none of the answer choices fall within this interval, we cannot determine the correct answer without additional information.
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a 5 pack of clay pots cost $14.25. What is the unit price?
Answers
Answer:4.25
Step-by-step explanation: you divide the cost by the materials I believe (I’m sorry if it’s wrong)
Answer:
2.85
Step-by-step explanation:
In order to find a unit price (price per item), you will need to divide.
In this case, we have the numbers 14.25 and 5.
So if we divide...
14.25 ÷ 5 = 2.85
We know the unit price.
The answer is 2.85.
Brainliest please?
6. Write the standard form of the line: y=2/5x-7
Answers
The standard form for a linear equation is: ax + by + c = 0
Once we move all the variables and constants to one side, we get the standard form for a linear equation
\(y = \frac{2}{5} x - 7\\y -\frac{2}{5}x = -7\\ y - \frac{2}{5}x + 7 = 0\\\)
Hope that helps!
what is multiplying polynomials calculator?
Answers
A multiplying polynomials calculator is a tool used to multiply two or more polynomials together.
A multiplying polynomials calculator is a tool used to multiply two or more polynomials together. It can be helpful for simplifying algebraic expressions, solving equations, and performing various calculations in mathematics and engineering.
The calculator takes the coefficients of each polynomial as input and uses the distributive property to multiply the terms of each polynomial together. It then combines like terms and arranges the result in descending order of degrees. Multiplying polynomials can be a tedious and error-prone process, especially for large polynomials, so using a calculator can save time and reduce the chances of errors. The multiplying polynomials calculator is commonly used by students, teachers, engineers, and anyone working with algebraic expressions.
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Mike received a check for $215. He deposits all the money into his bank account and his new balance is $2,184. What his balance before he made the deposit?
Answers
Answer:
2599
Step-by-step explanation:
2,814-215 = 2599
2184
- 215
———-
1969
1969+215=2184
Dilations about a point I need help asap please
Answers
The coordinates of the image of the dilation of the quadrilateral MNOP are;
M = (0, 4) M' = (0, 3)
N = (1, 2) N' = (2, -1)
O = (0, 0) O' = (0, -5)
P = (-1, 2) P' = (-2, -1)
The drawing of the figure with the vertices labeled, created with MS Excel is attached
What is a dilation transformation?A dilation transformation is one in which a geometric figure is resized to become smaller or larger.
The coordinates of the dilation of a point about the point (0, 5) can be found by first translating the quadrilateral MNOP with regards to the point (0, 5), such that the center is located at the origin as follows;
The coordinates of MNOP are; M(0, 4), N(1, 2), O(0, 0), and P(-1, 2)
The coordinates of the image following the translation are;
M' = M - (0, 5) = (0, -1)
N' = N - (0, 5) = (1, -3)
O' = O - (0, 5) = (0, -5)
P' = P - (0, 5) = (-1, -3)
The translated points are then dilated as follows;
The coordinates of the image following the dilation are;
M'' = 2 × M' = (0, -2)
N'' = 2 × N' = (2, -6)
O'' = 2 × O' = (0, -10
P'' = 2 × P' = (-2, -6)
The above image are translated to their initial position to get;
M''' = M'' + (0, 5) = (0, 3)
N''' = N'' + (0, 5) = (2, -1)
O''' = O'' + (0, 5) = (0, -5)
P''' = P'' + (0, 5) = (-2, -1)
Please find attached the diagram of the dilated image created with MS Excel
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which of the j values satisfy the following inequality? 10>j+5
choose all that apply
j=3
j=4
j=5
Answers
J=3
You can see that I can not equal or go over 10 so you plug both of them and it does satisfy the equation
Answer:
All of them
Step-by-step explanation:
Khan 7th Grade, Quiz 3
What is the local minimum value of the function? (Round answer to the
nearest hundredth)
g(x) = x4 – 5x² + 4
Answer here
Answers
State if the following Conjecture is True or False
Given for any a and b both elements of the Real numbers
Answers
Answer:
False
Step-by-step explanation:
The conjecture is also true if a and b are both negative.
Use the following sets to answer the question.
A={1,2,3,4,5}
B={5,6,7,8,9}
Which answer shows the intersection of A and B?
{1,2,3,4,5,6,7,8,9}
{1,2,3,4,6,7,8,9}
{2,4,8}
{5}
Answers
Answer:
5
Step-by-step explanation:
5 is in the middle between each set of numbers making it the intersecting number
5. Diego is 165 cm tall. Andre is 1.7 m tall. Who is taller, Diego or Andre? Explain your
reasoning.
Answers
Answer:
Andre is taller than Diego
Step-by-step explanation:
There are 100 cm in a meter. If we use this information and convert Diego's height to meters, we find that Diego is 1.65 m. This is a smaller number, and therefore, Diego is shorter than Andre.
Find the length of RA. A. 42 B. 84 C. 14 D. 7
Answers
Answer:
\(\large \boxed{\mathrm{B. \ 84}}\)
Step-by-step explanation:
\(LU\) bisects \(RU\) and \(UA\).
\(RU=UA\)
\(3m+21=6m\)
Solve for m.
Subtract 3m from both sides.
\(21=3m\)
Divide both sides by 3.
\(7=m\)
Calculate \(RA\).
\(RA=3m+21+6m\)
\(RA=9m+21\)
Put m = 7.
\(RA=9(7)+21\)
\(RA=63+21\)
\(RA=84\)
Answer:
B) 84
Step-by-step explanation:
ΔLRU ≅ ΔLAU {SAS congruent}
Therefore, UA = UR {CPCT}
6m = 3m +21
Subtract 3m from both sides
6m - 3m = 3m + 21 -3m
3m = 21
Divide both sides by 3
3m/3 = 21/3
m = 7
RA = RU + UA
= 3m + 21 + 6m {add like terms}
= 9m + 21 {Plug in m =7}
= 9*7 + 21
= 63 + 21
RA = 84 units
Penny reads 14 pages in 1/3 hour. What is the unit rate for pages per hour? For hours per page?
Answers
Answer: your
Pages per hour: 42
Hours per page: 1/42 of a hour
Step-by-step explanation:
14 x 3 = 42
1 hour divided by 42 pages = 1/42 of an hour per page
I was just wondering what 1/2-1/6 is as a fraction?
Answers
Another karaoke location, Sounds of the Sea, is charging $15 for 5 songs. Construct a graph to model the relationship between cost and number of songs.
Answers
The function that represent the proportional relationship between cost and number of songs is y = 3x and it's graph is attached below
Linear FunctionA linear function is a function that represents a straight line on the coordinate plane. For example, y = 3x - 2 represents a straight line on a coordinate plane and hence it represents a linear function. Since y can be replaced with f(x), this function can be written as f(x) = 3x - 2.
A linear function is of the form f(x) = mx + b where 'm' and 'b' are real numbers. Isn't it looking like the slope-intercept form of a line which is expressed as y = mx + b? Yes, this is because a linear function represents a line, i.e., its graph is a line. Here,
'm' is the slope of the line'b' is the y-intercept of the line'x' is the independent variable'y' (or f(x)) is the dependent variableTo find the relationship between cost and number of songs, we can write a proportional relationship for this
y = mx
m = constant of proportionality
15 = m(5)
m = 15 / 5
m = 3
The constant of proportionality is 3
The function to represent the relationship between the cost and number of songs is
y = 3x
The graph of the function is attached below
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Factorise fully 7xy + 21x I WILL GIVE BRANLIEST!
Answers
Answer:
7x(y + 3)
Step-by-step explanation:
\(7xy + 21x \\ \\ = 7x(y + 3)\)
The factorization of the given equation will be 7x(y + 3).
What is an equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
Given equation will be simplified as:-
= 7xy + 21x
= 7x ( y + 3 )
Therefore the factorization of the given equation will be 7x(y + 3).
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Calculate the area of the regular pentagon.
Answers
The area of the regular pentagon is 252.7m²
How to determine the valueThe formula that is used for calculating the area of a regular pentagon is expressed with the equation;
A = 5/2 x s x a
where the parameters are;
's' is the side of the pentagon 'a' is the apothem length.Apothem is the line from the center of the pentagon to a side, intersecting the side at 90 degrees right angle.
apothem,a = s/2 tan (180/n)
Substitute the values
Apothem, a = 7.6/2 tan 16
a = 7.6/0.57 = 13.3m
Substitute the values, we get;
Area = 5/2 × 7.6 × 13.3
Multiply the values, we have;
Area = 505. 4/2
Divide the values, we get;
Area = 252.7 m²
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The graph of which function will have a maximum and a y-intercept of 4?
f(x) = 4x2 + 6x – 1
f(x) = –4x2 + 8x + 5
f(x) = –x2 + 2x + 4
f(x) = x2 + 4x – 4
Answers
Answer:
its f(x) = –x^2 + 2x + 4
Step-by-step explanation:
\(f(x)=-x^2+2x+4\)
\(f(0)=-0^2 +2 x 0 +4\\f(0)=4\)
Answer:
C
Step-by-step explanation:
got it right on Edge
A square pyramid and its net are shown below. What is the surface area of the pyramid?
17 cm
16 cm
Type the answer in the box.
square centimeters
17 cm
16 cm
...15 sm
15 cm.
Answers
Check the picture below.
so the area of it, is really the area of a 16x16 square and four triangles with a base of 16 and a height of 15.
\(\stackrel{ \textit{\LARGE Areas}}{\stackrel{ square }{(16)(16)}~~ + ~~\stackrel{ \textit{four triangles} }{4\left[\cfrac{1}{2}(16)(15) \right]}}\implies 256~~ + ~~480\implies \text{\LARGE 736}~cm^2\)
