(c) (5 marks) Determine the MAP estimate of θ
(d) (5 marks) Determine the relative belief estimate of θ
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6. Suppose that the statistical model is given by fo(1) fo(2) fo(3) fol(4) θ=a11/3 1/6 1/3 1/6 0-...
f(x θ)-(1-0)0' i , X 1, 2, θ 6-(01) and 4.1 Consider one observation from the...
f(x θ)-(1-0)0' i , X 1, 2, θ 6-(01) and 4.1 Consider one observation from the p.d.f let the prior pd.f. λ on (0,1) be the 110, 1) distribution. Then, determine: (i) The posterior pdf. of θ, given x-x. ii) The Bayes estimate of 6, by using relation (15). The Bayes estimate Ολ(XI, , x.) defined in relation (14) can also be calculated thus (15) where h(θ , , x ) is the conditional p di of θ given X-X,...
Practice: Compute Likelihood of a Poisson Statistical Model 0/3 points (graded) Let X1,…,Xn∼iidPoiss(λ∗) for some unknown...
Practice: Compute Likelihood of a Poisson Statistical Model 0/3 points (graded) Let X1,…,Xn∼iidPoiss(λ∗) for some unknown λ∗∈(0,∞). You construct the associated statistical model (E,{Poiss(λ)}λ∈Θ) where E and Θ are defined as in the answers to the previous question. Suppose you observe two samples X1=1,X2=2. What is L2(1,2,λ)? Express your answer in terms of λ. L2(1,2,λ)= Next, you observe a third sample X3=3 that follows X1=1 and X2=2. What is L3(1,2,3,λ)? L3(1,2,3,λ)= Suppose your data arrives in a different order: X1=2,X2=3,X3=1....
4. Consider the following model: Let X = (1,X1,X2, Xs)". Suppose and that X is not perfectly colinear. Suppose further that ElY]oo and Exoo for 1 j S 3. Using a large sample of i.i.d. observa...
4. Consider the following model: Let X = (1,X1,X2, Xs)". Suppose and that X is not perfectly colinear. Suppose further that ElY]oo and Exoo for 1 j S 3. Using a large sample of i.i.d. observations from (XX1.x2, Xs), you estimate this equation using Ols. Let β-(%, 33) de- note the resulting estimate of β = (Ah-…β3). (a) Let θ-2A-β3. Is θη-2A-A3 a consistent estimate of θ? Is it an unbiased estimate of 0? (b) Express Var(0,] in terms of...
Number 4 turns out to be an inverse gamma function with parameters alpha= n and beta=...
Number 4 turns out to be an inverse gamma function with
parameters alpha= n and beta= the sum of x sub i
PLEASE ANSWER #5 NOT #4
4. Suppose that X1,X2, 10 pts. the p.d.f. is given by form a random sample from a distribution for which where the unknown parameter θ > 0. Suppose also that the improper prior of θ is m(0) Find the posterior distribution π(θ x). Hint: The inverse gamina distribution from question 6 in Homework...
For z e R and θ (0, 1), define otherwise. Let X1 , . . . , X" be i..d. random variables with density f, for some unknown θ E (0, 1) 1 point possible (graded, results hidden) To prepare, sketch th...
For z e R and θ (0, 1), define otherwise. Let X1 , . . . , X" be i..d. random variables with density f, for some unknown θ E (0, 1) 1 point possible (graded, results hidden) To prepare, sketch the pdf f, (z) for different values of θ E (0,1) Which of the following properties of fo (z) guarantee that it is a probability density? (Check all that apply) Note (added May 3) Note that you are not...
-3 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6...
-3 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 78 Re A sinusoid has an associated phasor shown in the graph above. Find the sinusoid's formula, in the form z t = Aco (2x100e十0) Enter the formula below. Give the phase value in radians between r(t) π and π cos(2100t+
As on the previous page, let X1,... ,Xn be iid with pdf where θ > 0. (to) 2 Possible points (qual...
As on the previous page, let X1,... ,Xn be iid with pdf where θ > 0. (to) 2 Possible points (qualifiable, hidden results) Assume we do not actually get to observe Xı , . . . , X. . Instead let Yı , . . . , Y, be our observations where Yi = 1 (Xi 0.5) . Our goal is to estimate 0 based on this new data. What distribution does Y follow? First, choose the type of distribution:...
solve by Excel Question 4 (6 marks) Suppose the compressive strength of batch of concrete is...
solve by Excel
Question 4 (6 marks) Suppose the compressive strength of batch of concrete is normally distributed with an unknown mean and known standard deviation of 4 MPa. Suppose 20 compressive strength tests are conducted and the arithmetic (or sample) mean is found to be 37.5 MPa. Plot the prior and posterior density functions for the mean strength of concrete for the following three priors: 1) Prior mean is 35 MPa and prior standard deviation is 5 MPa. 2)...
Concept Check: Terminology 0/3 points (graded) Suppose you observe iid samples X1,…,Xn∼P from some unknown distribution...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
8396 5101281 5 8 2 0 1 12 ( 4 2 1 ) ) ) 0000 f-000 0246802 (i) Defining fo-f(zo). Л that the quadratic f(x) and f2 f(x2), where Zo-x1-h and x2-xuth, show 2 , f2 - jo 2h2 2h is the quadratic interpol...
8396 5101281 5 8 2 0 1 12 ( 4 2 1 ) ) ) 0000 f-000 0246802 (i) Defining fo-f(zo). Л that the quadratic f(x) and f2 f(x2), where Zo-x1-h and x2-xuth, show 2 , f2 - jo 2h2 2h is the quadratic interpolating function for fo, fı and f2 (i.e. show that p(x)-f) 4] (ii) Use the interpolating polynomial p(x) as defined above, with Zo-12, xỉ-1.4 and 22 -1.6 (and fo, fı and f2 given by the table...
f(x θ)-(1-0)0' i , X 1, 2, θ 6-(01) and 4.1 Consider one observation from the p.d.f let the prior pd.f. λ on (0,1) be the 110, 1) distribution. Then, determine: (i) The posterior pdf. of θ, given x-x. ii) The Bayes estimate of 6, by using relation (15). The Bayes estimate Ολ(XI, , x.) defined in relation (14) can also be calculated thus (15) where h(θ , , x ) is the conditional p di of θ given X-X,...
4. Consider the following model: Let X = (1,X1,X2, Xs)". Suppose and that X is not perfectly colinear. Suppose further that ElY]oo and Exoo for 1 j S 3. Using a large sample of i.i.d. observations from (XX1.x2, Xs), you estimate this equation using Ols. Let β-(%, 33) de- note the resulting estimate of β = (Ah-…β3). (a) Let θ-2A-β3. Is θη-2A-A3 a consistent estimate of θ? Is it an unbiased estimate of 0? (b) Express Var(0,] in terms of...
Number 4 turns out to be an inverse gamma function with
parameters alpha= n and beta= the sum of x sub i
PLEASE ANSWER #5 NOT #4
4. Suppose that X1,X2, 10 pts. the p.d.f. is given by form a random sample from a distribution for which where the unknown parameter θ > 0. Suppose also that the improper prior of θ is m(0) Find the posterior distribution π(θ x). Hint: The inverse gamina distribution from question 6 in Homework...
For z e R and θ (0, 1), define otherwise. Let X1 , . . . , X" be i..d. random variables with density f, for some unknown θ E (0, 1) 1 point possible (graded, results hidden) To prepare, sketch the pdf f, (z) for different values of θ E (0,1) Which of the following properties of fo (z) guarantee that it is a probability density? (Check all that apply) Note (added May 3) Note that you are not...
-3 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 78 Re A sinusoid has an associated phasor shown in the graph above. Find the sinusoid's formula, in the form z t = Aco (2x100e十0) Enter the formula below. Give the phase value in radians between r(t) π and π cos(2100t+
As on the previous page, let X1,... ,Xn be iid with pdf where θ > 0. (to) 2 Possible points (qualifiable, hidden results) Assume we do not actually get to observe Xı , . . . , X. . Instead let Yı , . . . , Y, be our observations where Yi = 1 (Xi 0.5) . Our goal is to estimate 0 based on this new data. What distribution does Y follow? First, choose the type of distribution:...
solve by Excel
Question 4 (6 marks) Suppose the compressive strength of batch of concrete is normally distributed with an unknown mean and known standard deviation of 4 MPa. Suppose 20 compressive strength tests are conducted and the arithmetic (or sample) mean is found to be 37.5 MPa. Plot the prior and posterior density functions for the mean strength of concrete for the following three priors: 1) Prior mean is 35 MPa and prior standard deviation is 5 MPa. 2)...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
8396 5101281 5 8 2 0 1 12 ( 4 2 1 ) ) ) 0000 f-000 0246802 (i) Defining fo-f(zo). Л that the quadratic f(x) and f2 f(x2), where Zo-x1-h and x2-xuth, show 2 , f2 - jo 2h2 2h is the quadratic interpolating function for fo, fı and f2 (i.e. show that p(x)-f) 4] (ii) Use the interpolating polynomial p(x) as defined above, with Zo-12, xỉ-1.4 and 22 -1.6 (and fo, fı and f2 given by the table...
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