2. Let Zbe a Markov chain on a state space S = {1,… … , m} with transition probability matrixP(i, j) = Tij. Suppose that sample paths of Z are observed from which we recordNjj #i →→ j transitions of Z,mN₁ = [N₁; (# transitions out of state i).j=1It is known that the likelihood function of the observations of Z is given bymNijΠΠΠ.i=1_ j=1L =Incorporating the constraint j = 1 for each i = S,j=1m(a) Write the loglikelihood function of the observation.(b) Show that the maximum likelihood estimator of i → j transition probability ijisN=Tij =(c) Let m = 2 and assume that the number of transitions of Z is given by the matrixNijN₁50 10025 75Use the above information to derive estimators for 11, 12, 721 and 22.

(a)The log likelihood function can be written as:log L = sum(i, j) [N(i, j) * log(π(i, j))]Since we…

Answered: j) = ¡j. Suppose that sample paths of Z…

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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Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.
1a) Derive a maximum-liklihood estimator for the unknown parameter Y 1b) An experienced sales person completes sales following a Poisson distribution, with a mean rate of Y = 1.8 sales/month. A junior sales person completes sales at a mean rate of Y = 0.85 sales per month Find the probability of the joint sales team (experienced and junior) completing exactly 2 sales in any given months, assuming the two sales people act independently of each other. 1c) Find the probability of the joint sales team (experienced and junior) completing more than 2 sales in any given months
a) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION A
2) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function Where, ß > 0. If fram e ¹- {** f(x;0)=B for x > 4 otherwise "1 g) What is the maximum likelihood estimator of g (B) = 2(ß + 1). h) Is the maximum likelihood estimator of g (B) = 2(B + 1) unbiased or not?
Suppose model (XY, XZ, YZ) holds in a 2 x 2 x 2 table, and the common XY conditional log odds ratio at the two levels of Z is positive If the XY and YZ conditional log odds ratios are both positive or both negative, show that the XY marginal odds ratio is larger than the XY conditional odds ratio.
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
18. Show that for type lII population - x/0 dp O
a) Let X₁, X₂, X₁,...,X, be a random sample of size n from population X. Suppose that X-N(0, 1) and Y = -1-8√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y2 is between 0.02 and 5.02?
Please do question 2e and 2f with full handwritten working out
Compute the ff.: Expected Value of X ~ N(0, 1) Variance of X ~ N(0, 1) Moment Generating function of X ~ N(0, 1)
2) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.

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