Obtain the method of moments estimate of θ .

Question

Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)

Answer 1

Question

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

Chapter 8.10, Problem 4P

a.

To determine

Obtain the method of moments estimate of θ.

b.

To determine

Obtain the approximate standard error for the estimate.

b.

To determine

Obtain the approximate standard error for the estimate.

c.

To determine

Obtain the maximum likelihood estimate of θ.

d.

To determine

Find the approximate standard error of the maximum likelihood estimate.

e.

To determine

Find the posterior density and the mode of the posterior.

Plot the posterior density.

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Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)

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Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…

Answer 2

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Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

Chapter 8.10, Problem 4P

a.

To determine

Obtain the method of moments estimate of θ.

b.

To determine

Obtain the approximate standard error for the estimate.

b.

To determine

Obtain the approximate standard error for the estimate.

c.

To determine

Obtain the maximum likelihood estimate of θ.

d.

To determine

Find the approximate standard error of the maximum likelihood estimate.

e.

To determine

Find the posterior density and the mode of the posterior.

Plot the posterior density.

Expert Solution & Answer

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Answer 3

Question

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

Chapter 8.10, Problem 4P

a.

To determine

Obtain the method of moments estimate of θ.

b.

To determine

Obtain the approximate standard error for the estimate.

b.

To determine

Obtain the approximate standard error for the estimate.

c.

To determine

Obtain the maximum likelihood estimate of θ.

d.

To determine

Find the approximate standard error of the maximum likelihood estimate.

e.

To determine

Find the posterior density and the mode of the posterior.

Plot the posterior density.

Expert Solution & Answer

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Answer 4

Question

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

Chapter 8.10, Problem 4P

a.

To determine

Obtain the method of moments estimate of θ.

b.

To determine

Obtain the approximate standard error for the estimate.

b.

To determine

Obtain the approximate standard error for the estimate.

c.

To determine

Obtain the maximum likelihood estimate of θ.

d.

To determine

Find the approximate standard error of the maximum likelihood estimate.

e.

To determine

Find the posterior density and the mode of the posterior.

Plot the posterior density.

Expert Solution & Answer

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Answer 5

Chapter 8.10, Problem 4P

a.

To determine

Obtain the method of moments estimate of θ.

b.

To determine

Obtain the approximate standard error for the estimate.

b.

To determine

Obtain the approximate standard error for the estimate.

c.

To determine

Obtain the maximum likelihood estimate of θ.

d.

To determine

Find the approximate standard error of the maximum likelihood estimate.

e.

To determine

Find the posterior density and the mode of the posterior.

Plot the posterior density.

Answer 6

Chapter 8.10, Problem 4P

a.

To determine

Obtain the method of moments estimate of θ.

Answer 7

Chapter 8.10, Problem 4P

a.

To determine

Obtain the method of moments estimate of θ.

Answer 8

b.

To determine

Obtain the approximate standard error for the estimate.

Answer 9

b.

To determine

Obtain the approximate standard error for the estimate.

Answer 10

b.

To determine

Obtain the approximate standard error for the estimate.

Answer 11

b.

To determine

Obtain the approximate standard error for the estimate.

Answer 12

c.

To determine

Obtain the maximum likelihood estimate of θ.

Answer 13

c.

To determine

Obtain the maximum likelihood estimate of θ.

Answer 14

d.

To determine

Find the approximate standard error of the maximum likelihood estimate.

Answer 15

d.

To determine

Find the approximate standard error of the maximum likelihood estimate.

Answer 16

e.

To determine

Find the posterior density and the mode of the posterior.

Plot the posterior density.

Answer 17

e.

To determine

Find the posterior density and the mode of the posterior.

Plot the posterior density.

Answer 18

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Answer 29

Ch. 8.10 - Prob. 4P Ch. 8.10 - Prob. 59P

Obtain the method of moments estimate of θ .

Obtain the method of moments estimate of θ.

Obtain the approximate standard error for the estimate.

Obtain the approximate standard error for the estimate.

Obtain the maximum likelihood estimate of θ.

Find the approximate standard error of the maximum likelihood estimate.

Find the posterior density and the mode of the posterior. Plot the posterior density.

Chapter 8 Solutions

Find the posterior density and the mode of the posterior.

Plot the posterior density.