Problem 5Let X1, X2, ...,Xn be a random sample from a distribution with pdfT“ for r € (0, µ)|0 otherwisea+1f(r) =where a and u are positive constants.First, suppose we know that u = 1, but we don't know a and we want to estimate it.(i) Find the maximum likelihood estimator of a. Is the estimator consistent? If yes, show it. Isthe estimator biased? If yes, is the bias positive or negative?(ii) Let mi be the sample mean (m1 = X). Find a method of moments estimator of a which usesmı. Is the estimator consistent? If yes, show it. Is the estimator biased? If yes, is the biaspositive or negative?Second, suppose we know neither u nor a and we want to estimate both parameters.(iii) State the minimization problem of the maximum likelihood estimators of a and of u. (Youdon't need to find these estimators.)(iv) Let m2 be the sample second moment (m2 = E X?). Find method of moments estimatorsof a and of u which are functions of m1 and m2.

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Answered: Problem 5 Let X1, X2, ...,Xn be a…

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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Problem 1. A continuous random variable X is defined by f(x)=(3+x)^2/16 -3 ≤ x ≤ -1 =(6-2x^2)/16 -1 ≤ x ≤ 1 =(3-x^2)/16 -1 ≤ x ≤ 3 a)Verify that f(x) is density. b)Find the Mean
Question 2 Ignore the term "maximum likelihood"
Problem 7: Let X be a continuous random variable with the probability density for f(x) = 3x2 values of x in [0,1], and f(x) = 0 elsewhere. Compute the expected value and variance of X.
Problems 5 and 6 refer to the discrete random variables X and Y whose joint distribution is given in the following table, so P(X = 1 and Y = -1) = 1/4, P(X = 1 and Y = 1) = 0, etc. Problem 5: Compute the marginal distributions of X and Y, and use these to compute E(X), E(Y), Var(X), and V ar(Y). Problem 6: Compute Cov(X, Y) and the correlation ρ for the random variables X and Y. Are X and Y independent? Y= -1 Y =0 Y =1 X =1 1/4 1/8 0 X =2 1/16 1/16 1/8 X =3 1/16 1/16 1/4
Look at one of the marginal distributions described, and consider either value a success. Estimate the true population proportion for this value with a 95% confidence interval. (Hint: you may need to use the plus-four method if the sample size is too small.) We are now going to treat the sample as being two separate samples from two populations. One of the populations being compared is the population described in your conditional distribution (from problem 10). The other will be the rest of the sample. (An example is given below on how to form populations out of your Project 1 results.) Do these two populations have significantly different proportions successes? Use an appropriate significance test with. Two-Way Table: Dim Light Bright Light Total 4 9 13 Part A (first set of houses) 7 7 14 Part B (Second set of houses) 8 6 14 Part C (Third set of houses) Total: 19 Total: 22 Total: 41 Marginal Distribution: Homes with dim…
Problem 10 : Independent random variables {U, X, W} have variances V[U] = 1, V[X] = 2, V[W] = 3.Let Y = 2U + 3X + 4W. Find the standard deviation of Y.
Question 3 Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will be equal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their income will be y − L. The chance of a safe return is 50%. Now suppose that there are two identical merchants, A and B, who are both risk averse expected utility maximisers with utility of income given by u(y) = ln y. The income of each merchant will be 8 if their own ship returns and 2 if it sinks. As previously, the probability of a safe return is 50% for each ship. However, with probability p ≤ 1/2 both ships will return safely. With the same probability p both will sink. Finally, with the remaining probability, only one ship will return safely. (iv) Compute the increase in the utility of each merchant that they could achieve from pooling their incomes (as a function of p). How does the benefit of pooling depend on the probability p? Explain intuitively why this…
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9.1) Suppose X1, X2, and X3, denotes a random sample from the exponential distribution with density function shown in the image. a) Which of the above estimators are unbiased for θ? b) Among the unbiased estimators of θ, which has the smallest variance?
J 1 Problem 126. Let X and Y be discrete random variables with joint probability mass function pX,Y (x, y) = C/[(x + y − 1)(x + y)(x + y + 1)], x, y = 1, 2, 3, . . . Determine the marginal mass functions of X and Y
Problem 1. Consider the following density function. f(x )=[ (kx) ^ (2/3) * 0 < x < 2 Find the value of k. Find the cumulative distribution function ( CDF) of X Find the inverse of the CDF. Simulate a random sample of 10000 values from the above distribution by using inversetransformation and find the mean and the variance of those values, and write the Rcode.
If marginal denisties was given for two independent random variables f(x)=1/2 for 0<x<2 π(y)=1/3 for 0<y<3 Then find the value of P(X2 + Y2 > 1).

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