The probability density function given below describes a probability distribution used to model scores on certain...
The probability density function given below describes a probability distribution used to model scores on certain exams/tests:
?(?)={(?+1)?? for 0≤?≤1, 0 otherwise. The parameter θ must be greater than 1.
a. Find E(X).
A random sample of 10 test-takers gives the following scores in proportions:
0.96 0.43 0.77 0.85 0.93 0.79 0.77 0.85 0.74 0.98
b. Using part a, find the method of moments estimator for θ using the first moment of X based on the data above.
c. Find the maximum likelihood estimator for θ based on the data above.
Homework Answers
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The probability density function given below describes a
probability distribution used to model scores on certain...
Let X1, X2,.. Xn be a random sample from a distribution with probability density function f(z...
Let X1, X2,.. Xn be a random sample from a distribution with probability density function f(z | θ) = (g2 + θ) 2,0-1(1-2), 0<x<1.0>0 obtain a method of moments estimator for θ, θ. Calculate an estimate using this estimator when x! = 0.50. r2 = 0.75, хз = 0.85, x4= 0.25.
Please answer each of the parts in the following question. Show all steps. Thanks in advance!...
Please answer each of the parts in the following question. Show
all steps. Thanks in advance!
The following data represent the weights (in grams) of a random sample of 50 candies. 0.86 0.84 0.93 0.99 0.73 0.85 0.87 0.77 0.82 0.83 0.82 0.78 0.76 10.75 0.75 0.84 0.86 0.93 0.77 0.74 0.83 0.99 0.83 0.83 0.85 0.86 0.81 0.95 0.79 0.83 0.96 0.75 0.88 0.86 0.88 0.85 0.82 0.87 0.86 0.91 0.87 0.74 0.83 0.85 0.93 0.87 0.84 0.74 0.77...
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 X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random var...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density functior f(x)-oe ex , x > 0, θ > 0 Note that ElX] and VX]ー1 For the rest of this question, assume that you have a data set (xn1 consisting of a random sample of N observations of X. (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x;...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3e-dız?, x > 0. a. Find E(X), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for \, Gamma for the function, and pi for the mathematical constant 11. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/ I. Hint 1: Consider u = 1x2 or u = x2....
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...
Let X1, X2,.. Xn be a random sample from a distribution with probability density function f(z | θ) = (g2 + θ) 2,0-1(1-2), 0<x<1.0>0 obtain a method of moments estimator for θ, θ. Calculate an estimate using this estimator when x! = 0.50. r2 = 0.75, хз = 0.85, x4= 0.25.
Please answer each of the parts in the following question. Show
all steps. Thanks in advance!
The following data represent the weights (in grams) of a random sample of 50 candies. 0.86 0.84 0.93 0.99 0.73 0.85 0.87 0.77 0.82 0.83 0.82 0.78 0.76 10.75 0.75 0.84 0.86 0.93 0.77 0.74 0.83 0.99 0.83 0.83 0.85 0.86 0.81 0.95 0.79 0.83 0.96 0.75 0.88 0.86 0.88 0.85 0.82 0.87 0.86 0.91 0.87 0.74 0.83 0.85 0.93 0.87 0.84 0.74 0.77...
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 X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density functior f(x)-oe ex , x > 0, θ > 0 Note that ElX] and VX]ー1 For the rest of this question, assume that you have a data set (xn1 consisting of a random sample of N observations of X. (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3e-dız?, x > 0. a. Find E(X), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for \, Gamma for the function, and pi for the mathematical constant 11. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/ I. Hint 1: Consider u = 1x2 or u = x2....
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...
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