Show all working clearly. Thank you.
Show all working clearly. Thank you. 1. In this question, X is a continuous random variable...
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...
2. i) Let B be a random variable with the Binomial (n, p) distribution, so that Write down the likelihood function L(p) for m independent observations xi,...,Inm 2 marks 6 marks ili) Compute the bias and the mean squared error of the corresponding maximum likeli- from B. Int ii) Show that the maximum likelihood estimate for pis-Σ.ri. mn [7 marks] hood estimator of p. iv) Let X be a continuous random variable with density function for x > 0, and...
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
(b) Find the natural log of the likelihood function simplifying as much as possible. Loglikelihood = (c) Take the derivative of the log likelihood function you found in part (b) and make it 0. Solve for the unknown population parameter as a function of some of the summary statistics we know (X¯, or S 2 or whatever applies. ) That is your maximum likelihood estimator (MLE) of the unknown parameter. PART C ONLY Problem 2. Consider a random sample of...
Problem 3 variables with parameter Let r be an unknown constant. Let W be an exponential random A-1/3. Let Xr+w. (a) What is the maximum likelihood estimator of r based on a single observation X (b) What is the mean-squared error of the estimator from part (a):? (c) Is the estimator from part (a) biased or unbiased? Problem 3 variables with parameter Let r be an unknown constant. Let W be an exponential random A-1/3. Let Xr+w. (a) What is...
Please give detailed steps. Thank you. 5. Let {X, : i-1..n^ denote a random sample of size n from a population described by a random varaible X following a Poisson(θ) distribution with PDF given by θ and var(X) θ (i.e. you do not You may take it as given that E(X) need to show these) a. Recall that an estimator is efficient, if it satisfies 2 conditions: 2) it achieves the Cramer-Rao Lower Bound (CLRB) for unbiased estimators: Show that...
Let Xi,... , Xn be a random sample from a normal random variable X with E(X) 0 and var(X)-0, i.e., X ~N(0,0) (a) What is the pdf of X? (b) Find the likelihood function, L(0), and the log-likelihood function, e(0) (c) Find the maximun likelihood estimator of θ, θ (d) Is θ unbiased?
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...
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ