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Problem III. (12 points) Consider the following probability distribution. X 0 6 P(X = x) 1/4...
Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X =...
Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X = 1) 1/4 1/4 1/4 1/4 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1,...,xn) is proposed as ÔCX1,-- , Xx) = {x;I(X; # 0, and X; #6), n i=1 where I(-) is an indicator function, I(X; # 0, and X; #6) = 0, if X; = {0,6}, and I(X; # 0, and X; # 6) =...
Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4...
Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as n B(x,,,X,) X;I(Xi 70, and X; #6), n i=1 where I(-) is an indicator function, I(X; # 0, and Xi # 6) = 0, if X; € {0,6}, and I(X; # 0, and X; + 6) = 1, if Xi...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X =...
Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 1. (2 points) Find E(X). 2. (5 points) Find the sampling distribution of the sample mean à for samples of size n = 2.
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwi...
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwise, , where 0 > 0. Let X,i,.., X, be a random sample from this distribution. Not cavered 2011 (a) [6 marks] Find-4MM, the nethod of -moment estimator for θ for θ? If not, construct-an unbiased'estimator forg based on b) 8 marks Let X(n) n unbia estimator MM. CMM inbiase ( = max(X,, , Xn). Let 0- be another estimator of θ. 18θ...
Let X1, X2, ..., Xn be a random sample with probability density function a) Is ˜θ unbiased for θ? Explain. b) Is ˜θ consistent for θ? Explain. c) Find the limiting distribution of √ n( ˜θ − θ). need...
Let X1, X2, ..., Xn be a random sample with probability density
function
a) Is ˜θ unbiased for θ? Explain.
b) Is ˜θ consistent for θ? Explain.
c) Find the limiting distribution of √ n( ˜θ − θ).
need only C,D, and E
Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood fun...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where...
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
6. Suppose that X1, ..., Xn is a random sample from a population with the probability...
6. Suppose that X1, ..., Xn is a random sample from a population with the probability density function f(x;0), 0 E N. In this case, the esti- mator ÔLSE = arg min (X; – 6)? n DES2 i=1 is called the least square estimator of Ô. Now, suppose that X1, ..., Xn is a random sample from N(u, 1), u E R. Prove that the least square estimator of u is the same as maximum likelihood estimator of u.
Let X1, . . . , Xn be a random sample from a triangular probability distribution...
Let X1, . . . , Xn be a random sample from
a triangular probability distribution whose density function and
moments are:
fX(x) =
* I{0
x
b}
a. Find the mean µ of this probability
distribution.
b. Find the Method Of Moments estimator µ(hat) of µ.
c. Is µ(hat) unbiased?
d. Find the Median of this probability distribution.
I will thumbs up any portion or details of how to do this
problem, thanks!
We were unable to transcribe this...
Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X = 1) 1/4 1/4 1/4 1/4 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1,...,xn) is proposed as ÔCX1,-- , Xx) = {x;I(X; # 0, and X; #6), n i=1 where I(-) is an indicator function, I(X; # 0, and X; #6) = 0, if X; = {0,6}, and I(X; # 0, and X; # 6) =...
Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as n B(x,,,X,) X;I(Xi 70, and X; #6), n i=1 where I(-) is an indicator function, I(X; # 0, and Xi # 6) = 0, if X; € {0,6}, and I(X; # 0, and X; + 6) = 1, if Xi...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
Problem III. (12 points) Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 1. (2 points) Find E(X). 2. (5 points) Find the sampling distribution of the sample mean à for samples of size n = 2.
4. The Uniform (0,20) distribution has probability density function if 0 x 20 f (x) 20 0, otherwise, , where 0 > 0. Let X,i,.., X, be a random sample from this distribution. Not cavered 2011 (a) [6 marks] Find-4MM, the nethod of -moment estimator for θ for θ? If not, construct-an unbiased'estimator forg based on b) 8 marks Let X(n) n unbia estimator MM. CMM inbiase ( = max(X,, , Xn). Let 0- be another estimator of θ. 18θ...
Let X1, X2, ..., Xn be a random sample with probability density
function
a) Is ˜θ unbiased for θ? Explain.
b) Is ˜θ consistent for θ? Explain.
c) Find the limiting distribution of √ n( ˜θ − θ).
need only C,D, and E
Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
6. Suppose that X1, ..., Xn is a random sample from a population with the probability density function f(x;0), 0 E N. In this case, the esti- mator ÔLSE = arg min (X; – 6)? n DES2 i=1 is called the least square estimator of Ô. Now, suppose that X1, ..., Xn is a random sample from N(u, 1), u E R. Prove that the least square estimator of u is the same as maximum likelihood estimator of u.
Let X1, . . . , Xn be a random sample from
a triangular probability distribution whose density function and
moments are:
fX(x) =
* I{0
x
b}
a. Find the mean µ of this probability
distribution.
b. Find the Method Of Moments estimator µ(hat) of µ.
c. Is µ(hat) unbiased?
d. Find the Median of this probability distribution.
I will thumbs up any portion or details of how to do this
problem, thanks!
We were unable to transcribe this...
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