Suppose X i
∼N(0,a i
θ) independently for i=1,2,…,n where a i
(>0) are fixed and known constants for all i. Find the MLE of θ.
Answers
The maximum likelihood estimator (MLE) of θ is: = Σᵢ[1/(aᵢ²)] / Σᵢ[(xᵢ²)/(aᵢ²)]. To find the maximum likelihood estimator (MLE) of θ in the given scenario, we need to maximize the likelihood function. Let's denote the likelihood function as L(θ) and the observed values as x₁, x₂, ..., xₙ.
The likelihood function is the product of the probability density functions (PDFs) for each observation:
L(θ) = f(x₁; θ) × f(x₂; θ) × ... f(xₙ; θ)
where f(x; θ) is the PDF of a normal distribution with mean 0 and standard deviation aᵢθ.
Taking the natural logarithm of the likelihood function (log-likelihood) simplifies the calculations:
log L(θ) = log f(x₁; θ) + log f(x₂; θ) + ... + log f(xₙ; θ)
We can now substitute the PDF of the normal distribution into the log-likelihood equation:
log L(θ) = (θ) = log (1/( √(2 π)a ₁θ)) + log (1 /( √( 2π)a ₂ θ)) + ... + log (1/(√(2 π)a ₙθ))
= -n/2 l og (2 π) - (1/2) Σᵢ[log(a ᵢθ)] - Σᵢ[(xᵢ²)/(2a ᵢ² θ ²)
To find the maximum likelihood estimator (MLE) of θ, we differentiate the log-likelihood function with respect to θ and set it to zero:
d(log L(θ))/dθ = 0
Let's compute the derivative:
d(log L(θ))/dθ = - (1/2) Σᵢ[1/(aᵢθ)] + Σᵢ[(xᵢ²)/(aᵢ²θ³)]
To simplify further, we multiply the entire equation by 2θ³ and rearrange the terms:
0 = - Σᵢ[(θ²)/(aᵢ²)] + θ Σᵢ[(xᵢ²)/(aᵢ²)]
Multiplying through by aᵢ² gives us:
0 = - θ² Σᵢ[1/(aᵢ²)] + θ Σᵢ[(xᵢ²)/(aᵢ²)]
Now, we can solve for θ by factoring out θ:
0 = θ * (- θ Σᵢ[1/(aᵢ²)] + Σᵢ[(xᵢ²)/(aᵢ²)])
For this equation to hold, either θ = 0 (which is not feasible) or:
θ Σᵢ[(xᵢ²)/(aᵢ²)] = θ Σᵢ[1/(aᵢ²)]
Dividing both sides by θ gives:
Σᵢ[(xᵢ²)/(aᵢ²)] = Σᵢ[1/(aᵢ²)]
Finally, we can solve for θ:
θ = Σᵢ[1/(aᵢ²)] / Σᵢ[(xᵢ²)/(aᵢ²)]
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Related Questions
Analyze the diagram below and complete the instructions that follow.
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Given that ZCFE ZCFA, mZCFB = 68°, and mZEFD= 62°, find the measure of ZAFD.
Answers
The measure of ZAFD is 50 degrees.
Based on the given information:
ZCFE is congruent to ZCFA.
mZCFB = 68°.
mZEFD = 62°.
To find the measure of ZAFD, we can use the fact that angles in the same segment of a circle are equal.
Since ZCFE and ZCFA are congruent, angle ZCFE is equal to angle ZCFA.
Therefore, mZCFA = mZCFE.
Now, let's find the measure of angle ZCFE:
mZCFE = mZCFB + mZEFD (Angle Addition Postulate)
mZCFE = 68° + 62°
mZCFE = 130°
Since ZCFA is congruent to ZCFE, we can conclude that mZCFA is also equal to 130°.
Now, to find the measure of ZAFD, we need to consider the angles in the quadrilateral ZAFD.
The sum of the angles in a quadrilateral is always 360°.
mZAFD + mZAFB + mZBFD + mZBFA = 360°
We know that mZAFB and mZBFD are equal to 90° because they are right angles (angle AFB and angle BFD are right angles in the diagram).
Therefore, we can rewrite the equation as:
mZAFD + 90° + 90° + mZBFA = 360°
Simplifying:
mZAFD + 180° + mZBFA = 360°
Rearranging the equation:
mZAFD + mZBFA = 360° - 180°
mZAFD + mZBFA = 180°
We know that mZBFA is equal to mZCFA, which is 130°.
Substituting the known values into the equation:
mZAFD + 130° = 180°
Subtracting 130° from both sides:
mZAFD = 180° - 130°
mZAFD = 50°
Therefore, the measure of ZAFD is 50 degrees.
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slove of the quotient of 952 and 4
Answers
Answer:
238
Step-by-step explanation:
the quotient simply means the result of the division. when you divide 952 by 4, you get 238
6> 4x-6 or 4x - 6 > 22
Answers
Answer:
hello I'll explain
Step-by-step explanation:
you do it step by step choose one solve it and then next one.
\(4x - 6 > 22 \\ 4x > 28 \\ x > 7\)
then the next one
\(6 > 4x - 6 \\ 12 > 4x \\ 3 > x\)
so x<3 or x> 7
hope you understand
if you have any questions don't hesitate to ask
answer the question below to be featured in a Mr. Beast video!
Answers
Answer:
Yeah, theres no way
Step-by-step explanation:
lol
Answer:
imagine lol
Step-by-step explanation:
rugged cabin co. provides pre-made materials to build cabins. to ensure materials are in supply and ready for quick delivery, the company offers cabins in only three sizes. each cabin has rectangular floor plan where the length is equal to 5 feet more than twice the width. which expression represents the area, in square feet, for each cabin size?
a. 2w^2+5W,where w is the width
b. 2w^2+5,where w is the width
c. 2w^2,where w is the width
d. 10w^2,where w is the width
Answers
The carbinet has rectangular floor plan where the length is equal to 5 feet more than twice the width. The the expression that represents the area in square feet is Option B 2w²+5. ∧where w is the width
How to calculate Area of rectangle?The area of the rectangle is the floor space the rectangle can occupy at a given time.
The given parameters are
L= 2w+5
The width = w
Then area is given by the formula
A=L*W
A=2w*w+5
Simplify the expression you have
A=2w²+5 where w is the width
Therefore the area is b. 2w²+5, where w is the width
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Calculate the missing information for the installment loan that is being paid off early.
Answers
The computation illustrates that the rebate fraction is 0.18
How to compute the value?Given that : the total number of payments = 36
The number of payments made = 21
Remaining payments = 36-21 =15
We know that the sum of the digits for n payments
S = (n(n+1))/2
sum of the digits payments remaining = 15*(15+1)/2
= (15*16)/2
= 240/2
= 120
Sum of the digits number of payments = 36*(37)/2
= 1332
= 666
Rebate fraction = 120/666 =0.18
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Complete question
Calculate the missing information for the installment loan that is being paid off early.
Number of payment = 36
Payment made = 21
Payment remaining = ?
Rebate fraction = ?
Kendall has written her own recipe for pot roast. Her recipe says that a 4-pound roast should take 4
hours to cook. It also says that a 10-pound roast should take 10 hours to cook.
In this relationship, × represents the weight (in pounds) of the pot roast, and y represents the total
cooking time (in hours).
Answers
The equation that describes Kendall's recipe is y = x.
It appears that Kendall's recipe is suggesting a linear relationship between the weight of the pot roast and the cooking time required. To find the equation that describes this relationship, we can use two points on the line: (4, 4) and (10, 10).
Using the slope-intercept form of a linear equation (y = mx + b), we can solve for the slope (m) first:
m = (y2 - y1) / (x2 - x1)
m = (10 - 4) / (10 - 4)
m = 1
Now that we have the slope, we can solve for the y-intercept (b) by plugging in one of the points:
y = mx + b
4 = 1(4) + b
b = 0
Therefore, the equation that describes Kendall's recipe is:
y = x
In other words, the cooking time (y) is equal to the weight of the pot roast (x), assuming a constant rate of cooking per pound.
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One-third of the students in Mrs. Hayko's class walk to school. Of the students who do not walk to school, four-fifths take the bus.
a.) What fraction of the students in Mrs. Hayko's class take the bus to school?
b.) How many students might be there in her class?
Answers
Answer:
The possible number of students in Mrs. Hayko's class is limited to 15 or 30, as higher multiples of 15 would exceed the desired class size.
Step-by-step explanation:
a)
Let 'x' be the total number of students in Mrs. Hayko's class.
One-third of the students walk to school: (1/3)x.
The remaining students who do not walk to school: (2/3)x.
Four-fifths of the non-walking students take the bus: (4/5) * (2/3)x.
Simplify to find the fraction of students taking the bus: (8/15)x.
b)
Consider different values for 'x' to find a whole number of students taking the bus.
Start with a small number, such as x = 15.
Calculate the number of students taking the bus using (8/15)x.
If the result is a whole number, it's a possible class size.
Repeat with different values of 'x' until a whole number is obtained.
The possible number of students in Mrs. Hayko's class could be 15, 30, or any other multiple of 15.
in a random sample of 400 registered voters, 120 indicated they plan to vote for trump for president. determine a 95% confidence interval for the proportion of all the registered voters who will vote for trump.
Answers
The 95% confidence interval for the percentage of all eligible voters who intend to support Trump is (0.251, 0.349). If the true fraction does fall inside this range, we may say with 95% confidence.
We can use the following calculation to calculate a confidence interval for the percentage of all registered voters who intend to support Trump:
CI = p ± z\(\sqrt((p(1-p))/n)\)
where:
p is the proportion of the sample that will support Trump in the vote.
The critical value for the desired confidence level is z (95% confidence corresponds to a z-value of 1.96).
n is the sample size
When we enter the values from the issue, we get:
p = 120/400 = 0.3
z = 1.96
n = 400
CI = 0.3 ± 1.96sqrt((0.3(1-0.3))/400)
CI = 0.3 ± 0.049
CI = (0.251, 0.349)
Therefore, The 95% confidence interval for the percentage of all eligible voters who intend to support Trump is (0.251, 0.349). If the true fraction does fall inside this range, we may say with 95% confidence.
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a committee of 4 is to be selected from a group of 9 women and 7 men. what is the probability that 2 men and 2 women are selected?
Answers
The probability of selecting 2 men and 2 women from a group of 9 women and 7 men is 11%.
The probability of selecting 2 men and 2 women from a group of 9 women and 7 men is \($\frac{{7 \choose 2}{9 \choose 2}}{{16 \choose 4}}$\) which simplifies to\($\frac{63}{220}$\) or approximately 28.6%. The probability of selecting 2 men and 2 women from a group of 9 women and 7 men is 11%.
The probability of selecting 2 men and 2 women from a group of 9 women and 7 men can be calculated by using the formula for combination, nCr. In this case, we would be selecting 4 people from 16 individuals, so n = 16 and r = 4. The formula is nCr = n!/(r!(n-r)!). This can be simplified to 16C4 = 16!/ (4!(12!)). The answer is 1820/21168, which can be simplified to 86/792, or approximately 11%. Therefore, the probability of selecting 2 men and 2 women from a group of 9 women and 7 men is 11%.
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Mike is now 4 times older than al.if al was 2 years old 8 years ago how old is mike now?
Answers
Mike is 40 years old now.
Let Al's age be x years. According to the problem statement,
Al was 2 years old 8 years ago.
Therefore, Al's present age is:x + 8 years
Now, Mike is 4 times older than Al. That is, the age of Mike is 4(x + 8) years.
Mike is now 4 times older than Al, if Al was 2 years old 8 years ago, then Mike is 36 years old now.
In order to verify the solution, you can use a direct method of substitution.
If Al's present age is x + 8, then his age 8 years ago was x + 8 - 8 = x.
Since Al was 2 years old 8 years ago, we have:x = 2
Therefore, Al's present age is:x + 8 = 2 + 8 = 10 years
Now, Mike is 4 times older than Al.
Therefore, Mike's age is 4 × 10 = 40 years.
This is consistent with the original statement that "Mike is now 4 times older than Al if Al was 2 years old 8 years ago".Therefore, Mike is 40 years old now.
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HELP ASAP
6 divided by 200+ 40=
Answers
According to a U.S. news poll, 38% of students in the class of 2013 had done an internship during their time as an undergraduate student. Dana is interested in finding out whether students at her university had an internship rate that was higher than the national average. She obtained a list of 25 randomly selected students from the population of all students at her university by requesting this information from the university’s institutional research office. She collected the responses and calculated that the proportion for her university was 43%. Which one of the following statements about the z-test is correct?
It is not safe to use the z-test for p, since the sample is not a random sample from the entire population (or cannot be considered as one).
It is not safe to use the z-test for p, since n * (1 ? po) is not large enough.
It is safe to use the z-test for p.
It is not safe to use the z-test for p, since n * po is not large enough.
A national poll by the Institute for College Access and Success showed that in 69% of college graduates from public and nonprofit colleges in 2013 had student loan debt. Dr. Blackman wanted to find out if her public nonprofit university had a lower proportion of students who graduated with student loan debt in 2013.
For this survey, the null hypothesis was that the proportion of students with graduated with student loan debt equals 69% and the alternative hypothesis is that the proportion with student loan debt does not equal 69%. The significance level for this test was 0.05.
The results of the hypothesis test of the new survey showed a p-value of 0.039.
Which of the following statements is correct? Check all that apply.
The results were statistically significant.
The results were not statistically significant.
The null hypothesis should be rejected.
The null hypothesis should be accepted.
The null hypothesis cannot be rejected.
Answers
Answer: The null hypothesis should be accepted and The results were not statistically significant
Step-by-step explanation:
Calculate the value of c in the triangle
Answers
C^2 = (40)^2 + (9)^2
C^2 = 1600 + 81
C^2 = 1681
C = 41 cm
Please give me a brainliest answer
please help me with this, it’s due in an hour
Answers
#4. For every second, the hot air balloon increases height by 5 feet.
#5. X&Y are proportional because for every hour, 5 feet are gained. Thus, after 7 hours, 35 feet are gained.
Thats all i have time to do right now. Good luck :)
A medicine dropper releases about 1 liter of medicine in a day. How many milliliters of medicine would be released in an hour?
Answers
Answer:
41.7 ml milliliters of medicine would be released in an hour.
Step-by-step explanation:
1 day = 1 Liter
24 hours = 1000 ml
Now,
1000/24 = 41.7 ml
Thus, 41.7 ml milliliters of medicine would be released in an hour.
-TheUnknownScientist 72
Answer:
41.7 ml milliliters of medicine would be released in an hour.
Step-by-step explanation:
47/100+3/10+47/100=/100
Answers
Answer:
The total is 97/100
Step-by-step explanation:
47/100+3/100=50/100 (aka 1/2)
50/100+47/100=97/100
Answer:
47/100+3/10+47/10=97/100
¹
47
47
+3
__
97
STATISTICS
Which of the following are NOT continuous data?
a) Speed of a TGV express train
b) Volume of wine produced by French vineyards in one year
c) Time taken to drive from Paris to Lyon
d) Population of France
Answers
Answer:
A
Step-by-step explanation:
A is the most one wich makes sence
10. Set up and evaluate the definite integral for the area of the surface generated by revolving the curve a) (3 pts.)y= 6x 3+ 2x1 ,1≤x≤2, about the x-axis; b) (3 pts.) x= 4y−1,1≤y≤4, about the y-axis.
Answers
The definite integral for the area of the surface generated by revolving the curve y = 6x^3 + 2x about the x-axis, over the interval 1 ≤ x ≤ 2, can be set up and evaluated as follows:
∫[1 to 2] 2πy √(1 + (dy/dx)^2) dx
To calculate dy/dx, we differentiate the given equation:
dy/dx = 18x^2 + 2
Substituting this back into the integral, we have:
∫[1 to 2] 2π(6x^3 + 2x) √(1 + (18x^2 + 2)^2) dx
Evaluating this definite integral will provide the surface area generated by revolving the curve about the x-axis.
b) The definite integral for the area of the surface generated by revolving the curve x = 4y - 1 about the y-axis, over the interval 1 ≤ y ≤ 4, can be set up and evaluated as follows:
∫[1 to 4] 2πx √(1 + (dx/dy)^2) dy
To calculate dx/dy, we differentiate the given equation:
dx/dy = 4
Substituting this back into the integral, we have:
∫[1 to 4] 2π(4y - 1) √(1 + 4^2) dy
Evaluating this definite integral will provide the surface area generated by revolving the curve about the y-axis.
By setting up and evaluating the definite integrals for the given curves, we can find the surface areas generated by revolving them about the respective axes. The integration process involves finding the appropriate differentials and applying the fundamental principles of calculus.
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Graph the following features:
Slope = 4 Y-intercept: -1
Answers
Answer:
Graph the point at -1 on the y-axis, then the next at 4 on the x-axis, then connect them
This is the graph as in upload
english has 26 different letters. how many consecutive words have to be selected in a book to have at least 5 that start with the same letter?
Answers
The words need not be English language words 142506
What is permutation and combination?
A set of elements can be divided into subsets in two different ways: combination and permutation.
The components of the subset may be arranged in any order when combined. The components of the subset are listed in a permutation in a certain order.
If the word has 5 different letters then it is just 26-choose-5 or (26/5).
If the word has 4 different letters then it is just 26-choose-4 or (26/4)
have 4 options to choose which letter to duplicate twice.
This gives = 4*(26/4)
If the word has 3 different letters then it is just 26-choose-3 or (26/3).
you duplicate one of the three letters 3 times - 3 options for that, or you choose 2 letters and duplicate each one twice - again 3 options. In total this gives = 6*(26/3).
If the word has 2 different letters then it is just 26-choose-2 or (26/2).
The sets of 2 different letters and duplicate either one of the letters 4 times - 2 options; or duplicate one letter 2 times, and the other 3 times - again 2 options. In total this is = 4*(26/2).
The word has just 1 distinct letter you have = (26/1).
we get :
N = (26/5)+ 4*(26/4) + 6*(26/3) + 4*(26/2) + (26/1)
= 142506
The words need not be English language words = 142506
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Tony has 7 candy bars. he wants to split the candy bars into servings that are 1/3 of a bar. how many servings can he make?
Answers
uwuwuwuwuwuwuwuwuwuwuwuwuwuwu
7:1/3=7*3=21
What is the measure of the base of the rectangle if the area
of the triangle is 32 ft2 ?
Answers
Answer:
it would be a
Step-by-step explanation:
8x4=32
The base of the given triangle is calculated as: 16 ft
How to find the area of a triangle?The formula that is used to calculate the area of a triangle is expressed as:
Area of triangle= ½ × base × height
We are given that:
Area = 32 ft²
Height = 4 ft
Thus:
32 = ½ × base × 4
32 = 2 * base
base = 32/2
base = 16 ft
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X/19 = 4
what is x please help i don`t under stand this equation
p.s. xover 19 =4
Answers
Andrew Needs 995 tickets to get the prize he wants At the arcade. He Already has 319 tickets and can earn 4w tickets each time he plays an game. How many games will he have to play in order to have enough tickets to win the prize he wants.
Answers
To win the e prize
What are the correct trigonometric ratios that could be used to determine the length of ln? check all that apply. sin(20°) = startfraction l n over 8 endfraction cos(70°) = startfraction 8 over l n endfraction tan(70°) = startfraction l n over m n endfraction sin(20°) = startfraction 8 over l n endfraction cos(70°) = startfraction l n over 8 endfraction
Answers
The correct answer is option E which is Cos ( 70 ) = \(\dfrac{LN}{8}\).
The complete question is attached with the answer below:-
What is trigonometry?The branch of mathematics sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.
As we can see from the figure that ∠L = 70 and ∠M = 20 We can see that the length LN will be calculated as:-
Cos 70 = Base / Hypotenuse
So for an angle of 70, the base is LN and the hypotenuse is equal to 8 units.
Cos 70 = LN = 8
Therefore the correct answer is option E which is Cos ( 70 ) = \(\dfrac{LN}{8}\).
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Answer:
A & E
Step-by-step explanation:
Jus took it on edge
Johannes has saved $500.00 for a trip, and he plans on spending exactly $38.00 per day. If he wants to have at least $120.00 remaining at the end of his trip, find the greatest number of days his vacation can last.
Answers
Answer:No more than 10 Days
Step-by-step explanation:$500.00 - $120=$380/$38=10
A rectangular lawn is 100m long and 45m wide.
There are 3 circular ponds, with diameters of 20m, 10m and 5 especi Mrs Jones wants to cover the lawn with grass seed.
Each packet of grass seed covers 50m and costs £1. 49
How much will it cost Mrs Jones to cover the lawn with grass seed?
Answers
According to the area the amount needed to cover the lawn with grass seed is £121.8075
In math the term called area is defined as the total space occupied by the object.
Here we have know that the Length is 100m and the Breadth is 45m :-
Then the area is calculated as,
=> A = 100 × 45 = 4500m²
Here we have to do the calculation for 3 circular pond, then it can be calculated as,
=> 1st pond = 20/2 = 10m
=> 2nd pond = 10/2 = 5m
=> 3rd pond = 5/2 = 2.5m
Then the Area of 3 circular pond is calculated as,
=> A = (π × 10² + π × 5² + π × 2.5²)
Take the term π as common then we get,
=> A = π × ( 100 + 25 + 6.25)
Apply the value of π = 22/7, then we get,
=> A = 22/7 × (125 + 6.25)
When we simplify this one then we get
=> A = 22/7 × 131.25 = 2887.5/7
Therefore the area of the pond is 412.5m²
Now we need to do the calculation for required area for covering lawn, that can be written as
=> Required area = Area of lawn - Area of 3 circular ponds
Apply the value on it,
=> Required area = 4500 - 412.5 = 4087.5m²
As we know that packet of grass seed covers 50m, then it is simplified as,
=> Required Area = 4087.5/50 = 81.75m²
Now we need to do the calculation for cost of covering lawn is written as
=> Total cost = 81.75 × 1.49 = £121.8075
Therefore the cost of covering lawn with grass seed is £121.8075
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There are 134134 identical plastic chips numbered 11 through 134134 in a box. What is the probability of reaching into the box and randomly drawing a chip number that is greater than 5353
Answers
The probability of reaching into the box and randomly drawing a chip number greater than 5353 is approximately 0.9996 or 99.96%.
To calculate the probability of randomly drawing a chip number greater than 5353 from the box, we need to determine the total number of chips that meet this condition and divide it by the total number of chips in the box.
The total number of chips in the box is given as 134,134.
Now, we need to find the number of chips numbered 53,54,...,134,134, which is greater than 5353.
Since the chip numbers range from 11 to 134,134, the numbers 1 through 53 (5353 - 11) are excluded.
Therefore, the number of chips that meet the condition is:
Total number of chips - Number of excluded chips = 134,134 - 53 = 134,081
The probability of drawing a chip number greater than 5353 is then:
Probability = (Number of chips with numbers greater than 5353) / (Total number of chips)
Probability = 134,081 / 134,134 ≈ 0.9996
Therefore, the probability of reaching into the box and randomly drawing a chip number greater than 5353 is approximately 0.9996 or 99.96%.
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If angle 5 = 120°, what is angle 2?
Answers
Answer: 2=48
Step-by-step explanation:
5 divided by 120 is 24 2 divided by what is 24 = 48