(10) Student's t density with v degrees offreedom is given by f(x;v) =2HVTTv+12X 2Calculaterthe mean of a t distribution with v = 6degrees of freedom using a Metropolis -Hastings algorithm with the proposaldensity: (i) N(0,1); and (ii) t with v = 3degrees of freedom. (10) Student's t density with a degrees of freedom is given byf(2; 1) = (³)_1z(1 + z²/2}-{9423/2_Calculate the mean of a t distribution with = 6 degrees of freedom using a Metropolis-Hastings algorithm with the proposal density: (1) N(0,1); and (ii) ± with 2 = 3 degrees offreedom.

The Metropolis-Hastings algorithm is a Markov chain Monte Carlo (MCMC) method used for obtaining a…

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Q6. (a) Let X; (i = 1, ..,n) be independent random varables E(X;) = µ and var(X;) = of (i = 1, .,n). Let a = E-1 a; X; be an unbiased estimator of u where a; are constants. Find out how the a, are to be chosen if you wish to minimize var(@). (b) The bias and the mean squared error of an estimator Ô of a parameter 6 are defined as Bias (Ô) = b(ê ) = E(@) – 0 and m.s.e (@) = E{(@ – 0)?}, respectively. Show that m.s.e (@) var@) +{b(@ )}°.
The number of clients a store has during a week is Poisson distributed with - 25. The amount of money spent by each client follows a random = el,000t+250t2. Assuming the mean A variable with moment generating function o(t) independence between the number of clients, and the amount of money spent by each client. Find the mean and variance of the amount of money received by the store every week.
2) Let X be a continuous random variable with PDF 0
If the random variable X follows the uniform distribution U= (0,1) What is the distribution of the random variable Y= -2lnX. Show its limits.
Q3: Let X1. ,X, Binomail(1,0). (i) Show that B(1,0) is a member of the exponential class. (ii) Find the minimum variance unbiased estimator (MVUE) of 0.
2.3 Explain how to generate from the Weib(a, A) distribution using the inverse- transform method.
The quality characteristic of a product Y consists of two components X1, and X2 The distributions of these components are normal with means 30, 16 and variances 10, V(X2), respectively The specifications of the product are 101 10. If Y= 1X1 + -15X2. If it is required that Cpt is at least 1.2, find maximum V(X2) that would satisfy this requirement. Report the answer in 6 decimal places!
2 If X₁ and X₂ are the means of independent ran- dom samples of sizes n₁ and n₂ from a normal population with the mean u and the variance o2, show that the vari- ance of the unbiased estimator 2 w X₁ + (1-w). X₂ is a minimum when @ = n1 n₁ + n₂ idnu
Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.
(i) Let X,X2, be two identical independent distributed (iid)random variables,each Zx - 'X having chi – square distribution x2) with 2 degree of freedom.if Y, =: QI Y = X2, find: g(yy2). g(y,). (ii) Let x- gamma (4,B ), find the (mle) maximum liklihood estimator for B.
5. Find the moment generating function of the exponentially distributed random vari- able with pdf Xe-, x≥0 and use it to evaluate the mean and variance of X.
Suppose that the claim size distribution of an insurance portfolio follows a Pareto distribution of the form α+1 α f(x) β = B\B+x (i) Derive a formula for the rth moment, ar, of this Pareto distribution in terms of its (r-1) th moment, αr-1. Show your steps clearly with reasons. (ii) From this expression find α3 and α4 using the known result for the mean μ of the above Pareto distribution. (You may assume that α>r, where a is one of the two parameters of the Pareto distribution).

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(10) Student's / density with v degrees of freedom is given by f(x; r(+¹1) 2 ₁ V v+1 2 . Calculate (₂) VI the mean of at distribution with v = 6 degrees of freedom using a Metropolis - Hastings algorithm with the proposal density: (i) N(0, 1); and (ii) with v=3 degrees of freedom.