A random sample distribution of 6 observations (X1, X2, X3..., X6) is generated from a geometric(θ)...
A random sample distribution of 6 observations (X1, X2, X3..., X6) is generated from a geometric(θ) distribution, where θ ∈ (0, 1) unknown, but only T = Σ (from i=1 to 6) Xi is observed by the statistician.
a) describe the statistical model for the observed data (which is T)
b) is it possible to parameterize the model by Ψ = (1 - θ) / θ, prove your answer
c) is it possible to parameterize the model by Ψ = θ(1 - θ), prove your answer
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A random sample distribution of 6 observations (X1, X2, X3..., X6) is generated from a geometric(θ)...
6. Let Xi, X2, .., X6 be a random sample from a distribution with density function...
6. Let Xi, X2, .., X6 be a random sample from a distribution with density function 820-1 for 0 < x 1 where θ > 0 f(x; 6) 0 otherwise The null hypothesis Ho : θ-1 is to be rejected in favor of the alternative Ha : θ 1 if and only if at least 5 of the sample observations are larger than 0.7. What is the significance level of the test
Let X1,X2,X3,X4 be observations of a random sample of n-4 from the exponential distribution having mean...
Let X1,X2,X3,X4 be observations of a random sample of n-4 from the exponential distribution having mean 5, What is the mgf of Y-X1 X2 X3 X4? 4. 5. What is the distribution of Y? What is the mgf of the sample mean X = X+X+Xa+X1 ? 6. 7. What is the distribution of the sample mean?
6.1.12 Suppose that (x1,..., xn) is a sample from a Geometric(θ) distribution, where θ ∈ [0,...
6.1.12 Suppose that (x1,..., xn) is a sample from a Geometric(θ) distribution, where θ ∈ [0, 1] is unknown. Determine the likelihood function and a minimal sufficient statistic for this model. (Hint: Use the factorization theorem and maximize the logarithm of the likelihood.)
Question 6: [12 Marks: 5, 3, 41 Let X1, X2, ..., X6 be a random sample...
Question 6: [12 Marks: 5, 3, 41 Let X1, X2, ..., X6 be a random sample from a population following a Gamma distribution with parameters a and B. Consider the following two estimators of the mean (a/b) of this distribution. Ô2 = X And ôz = ž (X1 + X2 + X3) +ś (X4 + X5 + X3) Where I = (X1 + X2 + ... + X6) (a) Determine the sampling distribution of 7 using moment generating functions. (b)...
A sample of 4 observations (X1 = 0.4, X2 = 0.6, X3 = 0.7, X4 =...
A sample of 4 observations (X1 = 0.4, X2 = 0.6, X3 = 0.7, X4 = 0.9) is collected from a continuous distribution with pdf fx=θxθ-1, 0<x<1 (a) Find the point estimate of θ by the Method of Moments. (b) Find the point estimate of θ by the Method of Maximum Likelihood. Use two decimal places.
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. The observations are independent because
they were randomly drawn. Consider the following two point
estimators of the population mean μ:
1 = 0.10 X1 + 0.40
X2 + 0.40 X3 + 0.10
X4 and
2 = 0.20 X1 + 0.30
X2 + 0.30 X3 + 0.20
X4
Which of the following statements is true?
HINT: Use the definition of...
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. Consider the following estimator of
μ: 1 = 0.15 X1 +
0.35 X2 + 0.20 X3 + 0.30
X4. Using the linear combination of random
variables rule and the fact that X1, ...,
X4are independently drawn from the population, calculate
the variance of 1?
A.
0.55 σ2
B.
0.275 σ2
C.
0.125 σ2
D.
0.20 σ2
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function...
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function (0+1) A_1 fx(x) = fx(x; 0) = 20+1-xº(8 ?–1(8 - x), 0 < x < 8, 0> 0. a. Obtain the method of moments estimator of 8, 7. Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use mi for the sample mean X and m2 for the second moment. That is, m1 = 7 = + Xi, m2...
Exercise 5 Suppose that Xi, X2 X3 denote a random sample from a normal distribution with...
Exercise 5 Suppose that Xi, X2 X3 denote a random sample from a normal distribution with an unknown mean μ and a variance of 1. That is Xi ~ N(μ, σ-1). Consider two estimators, and μ2 9 For what values of μ doos μ2 obtain a lower MSE than μι, if any?
7.Let X1, X2, X3, and X4 be a random sample of observations from a population with...
7.Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ:⊝1 = 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4. Is this a biased estimator for the mean? What is the variance of the estimator? Can you find a more efficient estimator?
6. Let Xi, X2, .., X6 be a random sample from a distribution with density function 820-1 for 0 < x 1 where θ > 0 f(x; 6) 0 otherwise The null hypothesis Ho : θ-1 is to be rejected in favor of the alternative Ha : θ 1 if and only if at least 5 of the sample observations are larger than 0.7. What is the significance level of the test
Let X1,X2,X3,X4 be observations of a random sample of n-4 from the exponential distribution having mean 5, What is the mgf of Y-X1 X2 X3 X4? 4. 5. What is the distribution of Y? What is the mgf of the sample mean X = X+X+Xa+X1 ? 6. 7. What is the distribution of the sample mean?
Question 6: [12 Marks: 5, 3, 41 Let X1, X2, ..., X6 be a random sample from a population following a Gamma distribution with parameters a and B. Consider the following two estimators of the mean (a/b) of this distribution. Ô2 = X And ôz = ž (X1 + X2 + X3) +ś (X4 + X5 + X3) Where I = (X1 + X2 + ... + X6) (a) Determine the sampling distribution of 7 using moment generating functions. (b)...
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. The observations are independent because
they were randomly drawn. Consider the following two point
estimators of the population mean μ:
1 = 0.10 X1 + 0.40
X2 + 0.40 X3 + 0.10
X4 and
2 = 0.20 X1 + 0.30
X2 + 0.30 X3 + 0.20
X4
Which of the following statements is true?
HINT: Use the definition of...
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. Consider the following estimator of
μ: 1 = 0.15 X1 +
0.35 X2 + 0.20 X3 + 0.30
X4. Using the linear combination of random
variables rule and the fact that X1, ...,
X4are independently drawn from the population, calculate
the variance of 1?
A.
0.55 σ2
B.
0.275 σ2
C.
0.125 σ2
D.
0.20 σ2
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function (0+1) A_1 fx(x) = fx(x; 0) = 20+1-xº(8 ?–1(8 - x), 0 < x < 8, 0> 0. a. Obtain the method of moments estimator of 8, 7. Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use mi for the sample mean X and m2 for the second moment. That is, m1 = 7 = + Xi, m2...
Exercise 5 Suppose that Xi, X2 X3 denote a random sample from a normal distribution with an unknown mean μ and a variance of 1. That is Xi ~ N(μ, σ-1). Consider two estimators, and μ2 9 For what values of μ doos μ2 obtain a lower MSE than μι, if any?
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