2. Let X₁, Xn~ N(o, 2) be an iid Sample with unknown mean and variance.(a) Write down an expression for the log-likelihood(b) Compute the score function i.e.Tu(µ₁,0|x) = (₁²(14,0|x), (1,0|x))"до(c) By direct calculation show that expected value of the score function eval-uated at the true parameter is zero.(d) Compute the Fisher Information matrix for (u,0).(e) What would the observed information matrix be in this case? Does it differfrom the Fisher Information matrix?(f) Find the Fisher Information matrix for (u, o²).(g) The d-dimensional version of the Cramer-Rao Theorem states that anyunbiased estimator for a parameter has a variance that satisfies:uª Varo (Ô(X))u ≥ uª Ix (00)¯¹ufor any u € Rd.Using that result determine whether the unbiased estimator of o² given byis an efficient estimator.N1S² = Σ(x − x)2-n 1i=1

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Answered: 2. Let X₁,..., Xn~ N(o, o) be an iid…

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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If a discrete random variable X has the following probability distribution: X -2, - 1, 0, 1, 2 P(X) 0.2, 0.3, 0.15, 0.2, 0.15 Use this to find the following: (a) The mean of X and E[X^2]. (b) The probability distribution for Y = 2X^2 + 2 (i.e, all values of Y and P(Y )). (c) Using part (b) (i.e, the probability distribution forY ), find E[Y ]. (d) Using part (a), verify your answer in part (c) for E[Y ]. **Note: Please do not just copy from Chegg!
2. Cytomegalovirus (CMV) is a (generally benign) virus that infects one – half of young adults. If a random sample of 10 young adults is taken, find the probability that between 30% and 40% (inclusive) of those sampled will have CMV. Use the normal approximation with the continuity correction.
An electrochemical engineer has manufactured a new type of fuel cell (a type of battery)which has to undergo testing to prove its duration: the time it takes to go from fullycharged to completely uncharged, under a fixed nominal load. From the computational simulation models she has, the variance of the duration is σ2 = 4 h2 (hours squared)but she wants to estimate the mean duration time μ. To achieve this she is determinedto do the tests multiple times in independent but identical conditions. Can you findwhat is the smallest number of these tests that she has to do in order for her estimatedmean duration to be within ±0.2 h tolerance of the true mean with 95% certainty
A researcher that wanted to estimate the expectation AY of a random variable Y got three independent observations, Y, Y, Y The researcher knows the value o, of the variance of Y and is considering the following estimators: Pi = 4) Yi + (+) ¥ Py = () Yn + (;) ¥½ + () Y½ in (}) Yi + (}) ¥z + (() %D Which of the following is correct? ONone of the above Ois an unbiased estimator of l and it has the smallest variance of the three estimators. Ois an unbiased estimator of µ and it has the smallest variance of the three estimators. O and i, are both unbiased estimators of fl and Var (ſîz) < Var (îì3). is an unbiased estimator of µ and it has the smallest variance of the three estimators.
Let X1, X2, X3, ..., X, be a random sample from a distribution with known variance Var(X,) = o², and unknown mean EX, = 0. Find a (1 – a) confidence interval for 0. Assume that n is large.
let Y be a single observation from a normal distribution with mean θ and variance θ^2, θ>0.
X is distributed as a Normal random variable with mean of 100 and a standard deviation of 10 (i.e X~N(100,10). You take a random sample of 20 items from X and you calculate the average of this sample. What is the expected value of this sample average? (i.e. what value would you expect to get for the average of this small sample?
Q7) The length of time, in seconds, that a computer user takes to read his or her e- mail is distributed as lognormal random variable with μ = 1.8 and o² = 4. (a) (b) What is the probability that a user reads e-mail for more than 20 seconds? More than a minute? What is the probability that a user reads e-mail for a length of time that is equal to the mean of the underlying lognormal distribution?

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