Continue the calculations of Example 10.1.15, where the properties of the estimator of p(1 - p) were examined. (a) Show that, if p + 1/2, the MLE P(1 - ) is asymptotically efficient. (b) If p = 1/2, use Theorem 5.5.26 to find a limiting distribution of p(1-P). (c) Calculate the exact expression for Varſ (1-). Is the reason for the failure of the approximations any clearer?
Answers
(a) P(1-p) is asymptotically efficient when p+1/2.
(b) The limiting distribution of P(1-p) is 1/2 and variance 1/n.
(c) The exact expression for Var(P(1-p)) = 2p
How MLE P(1-p) is asymptotically efficient when p+1/2?(a) To show that the MLE P(1-p) is asymptotically efficient when p+1/2, we need to show that it achieves the Cramér-Rao lower bound (CRLB) on the variance of unbiased estimators. First, we calculate the derivative of the log-likelihood function:
ln L = n ln p + n ln (1-p)
ln L' = n/p - n/(1-p)
Taking the second derivative, we get:
ln L'' = -n/p² - n/(1-p)²
Now, we calculate the CRLB:
CRLB = [1/E(ln L'')] = [-E(ln L'')]' = [-n/p² - n/(1-p)²]'
CRLB = 2n/(p(1-p))³
Next, we calculate the variance of P(1-p):
Var(P(1-p)) = Var(P) + Var(1-p) - 2Cov(P, 1-p)
= p(1-p)/n + p(1-p)/n - 2Cov(P, 1-P)
Since P and 1-P are negatively correlated, Cov(P, 1-P) <= 0. Therefore,
Var(P(1-p)) >= 2p(1-p)/n
Dividing the CRLB by the variance of P(1-p), we get:
[CRLB]/[Var(P(1-p))] = (2n/(p(1-p))³) / [2p(1-p)/n] = n/(p(1-p))²
Since p+1/2, we have (p(1-p))² <= 1/4.
Therefore,
[CRLB]/[Var(P(1-p))] >= 4n
Thus, we have shown that P(1-p) is asymptotically efficient when p+1/2.
How to find a limiting distribution of p(1-P)?(b) If p = 1/2, we have:
Var(X) = p(1-p) = 1/4
E(X) = p = 1/2
According to Theorem 5.5.26, when n is large, the distribution of the MLE approaches a normal distribution with mean E(X) and variance 1/[nI(p)], where I(p) is the Fisher information.
We calculated I(p) in part (a) as:
I(p) = n/(p(1-p))²
Substituting p = 1/2, we get:
I(1/2) = 4n
Therefore, the limiting distribution of P(1-p) when p=1/2 is approximately normal with mean 1/2 and variance 1/[4n(1/2)(1/2)] = 1/n.
How to find the exact expression for Varſ (1-)?(c) To calculate the exact expression for Var(P(1-p)), we have:
Var(P(1-p)) = Var(P) + Var(1-p) - 2Cov(P, 1-p)
= p(1-p)/n + p(1-p)/n - 2Cov(P, 1-P)
= 2p(1-p)/n - 2Cov(P, 1-P)
To find Cov(P, 1-P), we use the fact that P and 1-P are binomially distributed with parameters n and p:
Cov(P, 1-P) = E(P(1-P)) - E(P)E(1-P)
= E(P) - E(P)² - E(P) + E(P)²
= -Var(P)
Therefore,
Var(P(1-p)) = 2p
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Related Questions
help pls his it due in 10 min
Answers
A. 5(2x+1)
10x+5
B. 34(x-1)
34x-34
C. 9(x+3)
9x+27
Hope it helps
For the function g(x) = 4x – 48, find g(-4).-4.)(Enter your answer as Ø if the function does not have a value at x =Provide your answer below:g{ – 4)=
Answers
Solution:
Given:
\(g(x)=\sqrt[3]{4x-48}\)g(-4) means the value of g(x) when x = -4
Substituting x = -4 into the function,
\(\begin{gathered} g(x)=\sqrt[3]{4x-48} \\ g(-4)=\sqrt[3]{4(-4)_{}-48} \\ g(-4)=\sqrt[3]{-16_{}-48} \\ g(-4)=\sqrt[3]{-64} \\ g(-4)=-4 \end{gathered}\)Also, showing the function on a graph using a graph plotter, the function is as plotted below;
The point when x = -4 corresponds to y = -4
Therefore,
\(g(-4)=-4\)
A plane leaves Atlanta's Hartsfield Airport and flies north for and west for at an average speed of . Find the bearing that the plane should take for the return trip. Round to the nearest tenth of a degree.
Answers
The plane should take a bearing of approximately 48.8 degrees for the return trip, which is rounded to the nearest tenth of a degree.
To determine the bearing for the return trip, we can use trigonometry and vector addition. Since the plane flies north and then west, we can consider the northward and westward components separately.
Let's assume the northward distance traveled is x and the westward distance traveled is y. The time taken for each leg of the trip is the same, so the ratio of the distances is equal to the ratio of the speeds.
Given that the average speed for the northward leg is 300 mph and the average speed for the westward leg is 400 mph, we have:
x / 300 = y / 400
Cross-multiplying, we get:
400x = 300y
To find the bearing, we need to find the tangent of the angle formed by the northward and westward components. The tangent is given by the ratio of the distances:
tan(θ) = y / x
Substituting the relationship between x and y, we have:
tan(θ) = 400 / 300
Using inverse tangent, we can find the angle θ:
\(\theta = tan ^{-1}(400 / 300)\)
Calculating this value, we find θ = 48.8 degrees.
Therefore, the plane should take a bearing of approximately 48.8 degrees for the return trip.
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Does anyone wanna trade in adopt me Imao or have like sum really good pets yall wanna give me cuz I’m amazing tehe anyways I’m u wanna my user is Matrix0425
Answers
Answer:
ill awnswer 4 that person that wants brainliest
Step-by-step explanation:
Answer:
no
Step-by-step explanation:
A lock has a code of 4 numbers between 1 and 20. If no numbers in the
code are allowed to repeat, how many different codes could be made?
Answers
There are 116,280 different codes that can be made with 4 numbers between 1 and 20 without repeating any number.
What are permutation and combination?In arithmetic, combination and permutation are two different ways of grouping elements of a set into subsets. In combination, the components of the subset can be recorded in any order. In a permutation, the components of the subset are listed in a distinctive order.
Here,
To find the number of different codes that can be made with 4 numbers between 1 and 20 without repeating any number, we can use the permutation formula, nPr = n! / (n-r)! where n is the total number of items and r is the number of items we want to choose.
In this case, we have n = 20 (the total number of numbers we can choose from) and r = 4 (the number of numbers we need to choose).
So, the number of different codes that can be made is:
20P4 = 20! / (20-4)! = 20! / 16! = 20 × 19 × 18 × 17 = 116,280
Therefore, there are 116,280 different codes that can be made with 4 numbers between 1 and 20 without repeating any number.
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The Roberts family is moving. They have storage tubs that are 2 ft long by 3 ft wide by 1.5 ft tall. What is the volume of these tubs?
please help :)??
Answers
Answer:
9
Step-by-step explanation:
Multiply all of them!
2*3=6
6*1.5=9
The answer is 9 :D
chris has been given a list of bands and asked to place a vote. his vote must have the names of his favorite and second favorite bands from the list. how many different votes are possible?
Answers
There are nC2 different votes possible, where n is the number of bands on the list and nC2 represents the number of ways to choose 2 bands out of n.
To calculate nC2, we can use the formula for combinations, which is given by n! / (2! * (n-2)!), where ! represents factorial.
Let's say there are m bands on the list. The number of ways to choose 2 bands out of m can be calculated as m! / (2! * (m-2)!). Simplifying this expression further, we get m * (m-1) / 2.
Therefore, the number of different votes possible is m * (m-1) / 2.
In the given scenario, we don't have the specific number of bands on the list, so we cannot provide an exact number of different votes. However, you can calculate it by substituting the appropriate value of m into the formula m * (m-1) / 2.
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You want to have $25,000 in your account after five years. Assuming you do not deposit or withdraw from the account, how much money should you deposit now if the savings account has a 6. 0% annual interest rate, compounded monthly?
Answers
The amount of money to be deposited now to have $25,000 in the account after five years is $18,580.65.
To find the amount of money to be deposited now to have $25,000 in the account after five years if the savings account has a 6.0% annual interest rate, compounded monthly, assuming that there is no deposit or withdrawal, we can use the formula for compound interest which is given by:
A=P(1+r/n)^nt
Where:
A is the future value, P is the present value, r is the annual interest rate, n is the number of times compounded in a year, and t is the number of years.
To solve the problem, we are to find P.
Thus, we can rewrite the formula above as:
P = A / (1+r/n)^nt
Here A = $25,000,
r = 6.0%,
n = 12 (compounded monthly),
t = 5 years.
Plugging in the values:
P = $25,000 / (1 + 0.06/12)^(12*5)P
P = $18,580.65
Therefore, To have $25,000 in your account after five years with the assumption to not deposit or withdraw from the account then the amount of money to be deposited now to have $25,000 in the account after five years is $18,580.65.
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the board of a major credit card company requires that the mean wait time for customers when they call customer service is at most 4.00 minutes. to make sure that the mean wait time is not exceeding the requirement, an assistant manager tracks the wait times of 42 randomly selected calls. the mean wait time was calculated to be 4.14 minutes. assuming the population standard deviation is 0.55 minutes, is there sufficient evidence to say that the mean wait time for customers is longer than 4.00 minutes with a 98% level of confidence?
Answers
This fails to reject the null Hypothesis.
What is z test?
If the data is distributed normally, a z test can be used to determine whether the means of two populations differ from one another. The z test statistic's value must be determined, together with the null hypothesis and alternative hypothesis setups, for this reason. On the z critical value, the decision criterion is based. The distribution graph is divided into acceptance and rejection zones during hypothesis testing by the z critical value. The null hypothesis can be rejected if the test statistic lies inside the rejection region; otherwise, it cannot.
a recent survey reported that small businesses spend 24 hours a week marketing their business.
these businesses are spending less than 24 hours a week on marketing. the chamber conducts a survey of 58 small businesses within their state and finds that the average amount of time spent on marketing is 22.7 hours a week.
assuming that the population standard deviation is 6.1 hours.
z = x'- u / α(/√n) = (5.05-4.50/1.83/√38) = 1.85
Hence, This fails to reject the null Hypothesis.
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a player rolls two fair six sided dice until the player rolls doubles. the probability of rolling doubles with one roll of two fair dice is 1/6, what is the probability that it takes 3 rolls until the player rolls doubles
Answers
The probability that it takes three rolls until the player rolls doubles is 25/216.
The probability of rolling doubles with one roll of two fair six-sided dice is 1/6.
The probability of an event is nothing but a number between 0 and 1, where zero shows the impossibility, and one shows the certainty.
First, determine the probability of rolls to obtain doubles in each roll.
Fails in the first roll:
P = 1 - 1/6 = 5/6
Fails in the second roll:
P = 1 - 1/6 = 5/6
In the third roll, the player rolls doubles, that is,
p = 1/6
Now, the probability of requiring 3 rolls until getting doubles will be:
= 5/6 × 5/6 × 1/6
= 25/216
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When analyzing two quantitative variables, what is the first thing that should be done?.
Answers
Using the theories of quantitative variables, we got that Graphical display is the first thing that should be done when we are analyzing two quantitative variables.
Comparing the Two Quantitative Variables As we did when considering only one variable, we begin with the graphical display. A scatterplot is the most useful display technique for comparing the two quantitative variables. We plot on the y-axis the variable we consider for the response variable and on the x-axis we place the explanatory or the predictor variable.
Hence, When analyzing two quantitative variables, the first thing that should be done is Graphical display.
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a group of children held a grape-eating contest. when the contest was over, the winner had eaten grapes, and the child in -th place had eaten grapes. the total number of grapes eaten in the contest was . find the smallest possible value of . a group of children held a grape-eating contest. when the contest was over, the winner had eaten grapes, and the child in -th place had eaten grapes. the total number of grapes eaten in the contest was . find the smallest possible value of .
Answers
The smallest possible value of the total number of grapes eaten in the contest is the product of (n-1) and the number of grapes eaten by the child in last place, plus the number of grapes eaten by the winner.
The smallest possible value of the total number of grapes eaten in the contest is (n-1)*(grapes eaten by the child in last place) + (grapes eaten by the winner). So for example, if the winner had eaten 10 grapes and the child in last place had eaten 5 grapes, the total number of grapes eaten in the contest would be (n-1)*5 + 10 = 45.
The smallest possible value of the total number of grapes eaten in the contest is the product of (n-1) and the number of grapes eaten by the child in last place, plus the number of grapes eaten by the winner.
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What is the probability that , in a group of 200 random people , at least two of them have the same triple of initials (such as JTK for James Tiberius Kirk)
Answers
The probability is P(X >= 2).
200 random people, n = 200
Let us consider the Binomial expression
i.e.,
P(X = r) = nCr*p^r*q^(n-r)
Here we have to assume that the 3 initials are equally likely events.
i.e.,
p = 1/3
q = 1 - 1/3
= 2/3
Required probability = P(X >= 2)
= 1 - [P(X = 0) + P(X = 1)]
= 1 - [200C0*(1/3)^0*(2/3)^200 + 200C1*(1/3)^1*(2/3)^199]
= 1 - [1*1*(2/3)^200 + 200*(1/3)*(2/3)^199]
= 1 - [(2/3)^200 + 200*(1/3)*(2/3)^199]
= 1 - (2/3)^200 - 200*(1/3)*(2/3)^199
Probability is truly how possibly something is to happen. every time we're unsure about the final results of an event, we will talk about the possibilities of certain consequences and how probably they are. The evaluation of events ruled by means of possibility is called records.
3 styles of probability
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1. Which of the following is INCORRECT:
Independent random samples arise when ...
a. one random sample is split into groups differing by an observed feature
b. the individuals in a sample are randomly assigned to experimental groups
c. data is recorded repeatedly on a random sample of individuals
d. random samples are selected separately
2. The margin of error of a confidence interval about the difference between the means of two populations is equal to
a. half the width of the confidence interval
b. twice the width of the confidence interval
c. the width of the confidence interval
d. 1.5 times the width of the confidence interval
Answers
1. Independent random samples arise when one random sample is split into groups differing by an observed feature is incorrect.
2. The margin of error of a confidence interval about the difference between the means of two populations is equal to half the width of the confidence interval.
1. Independent random samples arise when individuals in a sample are randomly assigned to experimental groups, data is recorded repeatedly on a random sample of individuals, or random samples are selected separately. The statement that one random sample is split into groups differing by an observed feature does not accurately describe independent random samples.
2. The margin of error in a confidence interval represents the range of values within which the true population parameter is likely to fall. It is calculated by taking half of the width of the confidence interval. Therefore, the correct answer is that the margin of error is equal to half the width of the confidence interval.
In summary, the incorrect statement is that independent random samples arise when one random sample is split into groups differing by an observed feature. The margin of error of a confidence interval about the difference between the means of two populations is equal to half the width of the confidence interval.
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Part 1 of 2 Question content area top Part 1 Two of the most expensive cars in the world are car A and car B. The prices of these two cars differ by more than $. The price of car A is $. a. Assuming that you do not know which model is more expensive, write an absolute value inequality that describes this situation. Use x for the price of car B. b. What are the possibilities for the price of car B?
Answers
Considering the given situation, it is found that:
a. The absolute value inequality is: |x - 131745| > 15000.
b. The possibilities for the price of car B are of x < $116,745 or x > x > $146,745.
What is the absolute value function?The absolute value function is defined by the following piecewise rule, depending on the input of the function:
|x| = x, x ≥ 0.|x| = -x, x < 0.It gives the distance of a point x to the origin, hence, for example, |-2| = |2| = 0.
In this problem, the difference between the price of car B of x and the price of car A of $131,745 is greater than $15,000, hence the absolute value inequality that models this situation is:
|x - 131745| > 15000.
Hence the possibilities for the price of car B are:
x - 131745 < -15000 -> x < -15000 + 131745 -> x < $116,745.x - 131745 > 15000 -> x > 131745 + 15000 -> x > $146,745.What is the missing information?The complete problem is given by:
"Two of the most expensive cars in the world are car A and car B. The prices of these two cars differ by more than $15,000. The price of car A is $131,745. a. Assuming that you do not know which model is more expensive, write an absolute value inequality that describes this situation. Use x for the cost of car B. b. What are the possibilities for the price of car B?"
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y=-3x+12 find the intercepts of x and y
Answers
Answer: x intercept: (4,0) y intercept: (0,12)
Step-by-step explanation:
To find the x intercept, substitute in 0 for y and solve for x. To find the y intercept, substitute in 0 for x and alive for x.
What type of adaptation is a prehensile tail?
Answers
Prehensile tail in animals like monkeys or tree porcupine is a structural adaptation as it is a physical trait that help them to survive in their environment. The term prehensile means "able to grasp".
A prehensile tail's special characteristic is that it is used to grasp objects. A well defined prehensile tail can be used to grasp or hold objects helping the animal to finding and eating food.
A partially prehensile tail helps the organism to anchor itself to tree aiding in climbing. It is seen in tree porcupines, rats, many snakes etc. It is not seen only in mammals, even seahorses have prehensile tail.
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Use the quadratic formula to find the solutions to the quadratic equation
below.
3x^2-5x+4=0
Answers
Answer: x
=
5
±
i
√
23
6
Explanation:
y
=
3
x
2
−
5
x
+
4
=
0
Use the improved quadratic formula.
D
=
d
2
=
b
2
−
4
a
c
=
25
−
48
=
−
23
=
23
i
2
-->
d
=
±
i
√
23
There are no real roots. There are 2 complex roots:
x
=
−
b
2
a
±
d
2
a
=
5
6
±
i
√
23
6
=
5
±
i
√
23
6
Step-by-step explanation:
Answer:
X= (5 ± i√23)/6
fast please! Which measure of center is best to use for this set of data?
45, 39, 49, 37, 3, 51
Question 1 options:
Mean
Median
Mode
Range
Answers
Answer:
Median
Step-by-step explanation:
Mean: Add up all the numbers, then divide by how many numbers there are.
Median: Arrange your numbers in numerical order. Count how many numbers you have. Go to the number in that position and average it with the number in the next higher position to get the median.
Mode: Put the numbers in order from least to greatest and count how many times each number occurs.
Range: The difference between the smallest and highest numbers
Answer:
Mean:
\(mean = \frac{(45 + 39 + 49 + 37 + 3 + 51)}{6} \\ = 37.3\)
Median:
\(3 \: \: 37 \: \: 39 \: \: 45 \: \: 49 \: \: 51 \\ median = ( \frac{n}{2} + 1) {}^{th} \\ = ( \frac{6}{2} + 1) {}^{th} \\ = 4 {}^{th} \: value \\ median = 45\)
Mode:
\(mode = none \\ all \: digits \: are \: repeated \: once\)
Range:
\(range = 51 - 3 \\ = 48\)
Solve the following quadratic by factoring. x² + 3x – 40 = 0
A. X = -8, x=5
B. X= 8, x=5
C. X=8,x= -5
Answers
Answer:
That would be A!
Step-by-step explanation:
Please tell me if i made a mistake or misunderstood!
Factor -2x^3-4x^2-6x
Answers
Answer:
-2x(x^2+2x+3)
Step-by-step explanation:
-2x^3-4x^2-6x
You can only factor out the 2x the other part is not factorable.
Jimmy's Bakery sold 950 donuts last week. This week, the number of donuts sold increased 12%. How many more donuts were sold this week?
Answers
Answer:
114
Step-by-step explanation:
12% of 950 is 114, so since 12% more donuts are sold this week, we sold 114 more donuts this week compared to last week.
how many 2 digit numbers can be formed using only the digits 2, 3, 5, and 6, if the digits are not be repeated within a number
Answers
25
26
32
35
36
52
53
56
62
63
65
Answer:
12
Step-by-step explanation:
You want the number of two digit numbers that can be made from the digits {2, 3, 5, 6} if there are no repeats.
PermutationsThe first digit can be any of the four.
The second digit can be any of the remaining three, so there are ...
4×3 = 12 possible 2-digit numbers
__
Additional comment
They are ...
23, 25, 26, 32, 35, 36, 52, 53, 56, 62, 63, 65
The base of a triangle is 20m and its area is 250m.What is the height of the triangle
Answers
Answer:
the height of the triangle is 25
Answer:
height = 25 m
Step-by-step explanation:
The area (A) of a triangle is calculated as
A = \(\frac{1}{2}\) bh ( b is the base and h the height )
Here A = 250 and b = 20 , then
\(\frac{1}{2}\) × 20 × h = 250 , that is
10h = 250 ( divide both sides by 10 )
h = 25 m
can someone help me please!!!
Answers
Answer:
\(5x + 45 + 3x - 25 = 180\)
another equation
y=5x+45
given any set of seven integers, must there be at least two that have the same remainder when divided by 6?
Answers
Given any set of seven numbers that should be at least 2 that must have the same remainder when divided by 6
What are the factors?A factor is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the product since the product is divisible by them.
How do you find factors?Step 1: factor the provided number into primes, or describe it as the product of primes.
Step 2: Write the prime factorization in exponent form in step three.
Step 3: Increase each exponent by one.
Step 4: Multiply each of the results.
There can only be a maximum of six different remainders when dividing by six: 0, 1, 2, 3, 4, and 5.
In any set of seven integers, two must have the same remainder when divided by seven according to the pigeon hole principle.
b) Consider set of integers 0, 1 ,2, 3, 4, 5 and 66. All these have different remainders upon division by 8 and hence the answer is no.
The interpretation is yes
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How many solutions does this linear system have?
Y= -1/2x + 4 x + 2y=-8
Answers
Answer:
No solutions
Step-by-step explanation:
\(y=-\dfrac{1}{2}x+4 \\\\x+2y=-8\)
Substitute:
\(x+2(-\dfrac{1}{2}x+4)=-8\\\\x-x+8=-8\\\\8=-8\)
This system has no solutions.
Hope this helps!
I WILL GIVE BRAINLIEST!!!!
Answers
Answer:
c
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
How much greater is the value of the 6 inand 4786.53 Denton 3821.69
Answers
Jane wants to buy a CD for $12. She has saved $7. How much money does she still need if 6% sales tax will be added to the cost of the CD.
Answers
Using the sales tax, the amount of money she still needs is $5.72
What is sales tax?A sales tax is a consumption tax imposed by the government on the sale of goods and services.
Jane wants to buy a CD for $12. She has saved $7.
The amount she needs if 6% sales tax is added to the cost can be calculated as follows;
Let's find the sales tax,
sales tax = 6% of 12
sales tax = 6 / 100 × 12
sales tax = 72 / 100
sales tax = $0.72
Therefore,
amount she needs to save = 12 + 0.72 - 7
amount she needs to save = 12.72 - 7
amount she needs to save = $5.72
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