Homework Answers
The given PDF is 
a) The condition for PDF is
b) The expected value is
The second moment is
The method of moment estimator is found as
c) The moment estimator is unbiased if 
d) Here 
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You are given the following probability density function, φ2(x), for the cosine of the surface an...
You are given the following probability density function, 6x(r), for the cosine of the surface an...
You are given the following probability density function, 6x(r), for the cosine of the surface angle, S, of a laser etching tool. The distribution function has one parameter, a, and one constant, c. -1sxs1 a) What is the value of the constant, e? b) What is the moment estimator for a? c) Explain how you can determine if this moment estimator is unbiased. t... . 24) denote a random sample of sample size n 24 with sample mean of-0.01 and...
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...
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...
Moment of method estimation
Suppose a random sample X1, X2, ..., Xn is drawn from a distribution believed to
have the following probability density function:
Find the first moment of X supposing that the parameter α is known. Use it to find the
method of moment estimator for unknown parameter β. You may find it easier to use the
notation m1, m2, ..., mk to denote the first sample moment, second sample moment, ..., kthsample moment, respectively. Is the estimator you derived unbiased?
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.
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...
Let X1, X2, ..., Xn be a random sample of size n from the distribution with probability density function f(x;) = 2xAe-d...
Let X1, X2, ..., Xn be a random sample of size n from the distribution with probability density function f(x;) = 2xAe-de?, x > 0, 1 > 0. a. Obtain the maximum likelihood estimator of 1. Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use mi for the sample mean X, m2 for the second moment and pi for the constant 1. That is, n mi =#= xi, m2 = Š X?. For example,...
Show all working clearly. Thank you. 1. In this question, X is a continuous random variable...
Show all working clearly. Thank you.
1. In this question, X is a continuous random variable with density function (x)a otherwise where ? is an unknown parameter which is strictly positive. You wish to estimate ? using observations X1 , . …x" of an independent random sample XI…·X" from X Write down the likelihood function L(a), simplifying your answer as much as possi- ble 2 marks] i) Show that the derivative of the log likelihood function (a) is 4 marks]...
B2. (a) Suppose θ is an unknown parameter which is to be estimated from a single measurement X, distributed according to some probability density function f(r0). The Fisher information I(0) is define...
B2. (a) Suppose θ is an unknown parameter which is to be estimated from a single measurement X, distributed according to some probability density function f(r0). The Fisher information I(0) is defined by de Show that, under some suitable regularity conditions, the variance of any unbi- ased estimator θ of θ is then bounded by the reciprocal of the Fisher information Var | θ 1(8) Note that the suitable regularity conditions, which are not specified here, allow the interchange of...
Let with Y, Y, ..., Yn be i id random variables the following probability density function,...
Let with Y, Y, ..., Yn be i id random variables the following probability density function, 1 x)/x fyly) = f I y ocyc1 o otherwise a) b) where x>0 is an unknown parameter. Find the maximum likelihood estimator , ã of x. Show this is an unbaised estimator for a. Hint : make use of the fact that in y follows an exponential distribution with mean a. Toe., -lny ~ Exp(x) c) Find the MSE of the manimum likelihood...
You are given the following probability density function, 6x(r), for the cosine of the surface angle, S, of a laser etching tool. The distribution function has one parameter, a, and one constant, c. -1sxs1 a) What is the value of the constant, e? b) What is the moment estimator for a? c) Explain how you can determine if this moment estimator is unbiased. t... . 24) denote a random sample of sample size n 24 with sample mean of-0.01 and...
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...
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 of size n from the distribution with probability density function f(x;) = 2xAe-de?, x > 0, 1 > 0. a. Obtain the maximum likelihood estimator of 1. Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use mi for the sample mean X, m2 for the second moment and pi for the constant 1. That is, n mi =#= xi, m2 = Š X?. For example,...
Show all working clearly. Thank you.
1. In this question, X is a continuous random variable with density function (x)a otherwise where ? is an unknown parameter which is strictly positive. You wish to estimate ? using observations X1 , . …x" of an independent random sample XI…·X" from X Write down the likelihood function L(a), simplifying your answer as much as possi- ble 2 marks] i) Show that the derivative of the log likelihood function (a) is 4 marks]...
B2. (a) Suppose θ is an unknown parameter which is to be estimated from a single measurement X, distributed according to some probability density function f(r0). The Fisher information I(0) is defined by de Show that, under some suitable regularity conditions, the variance of any unbi- ased estimator θ of θ is then bounded by the reciprocal of the Fisher information Var | θ 1(8) Note that the suitable regularity conditions, which are not specified here, allow the interchange of...
Let with Y, Y, ..., Yn be i id random variables the following probability density function, 1 x)/x fyly) = f I y ocyc1 o otherwise a) b) where x>0 is an unknown parameter. Find the maximum likelihood estimator , ã of x. Show this is an unbaised estimator for a. Hint : make use of the fact that in y follows an exponential distribution with mean a. Toe., -lny ~ Exp(x) c) Find the MSE of the manimum likelihood...
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