Let Y₁, ₂,.. density function Yn denote a random sample from a uniform distribution with " For testing Ho: 0 = 0o vs. (LRT) gives RR = {Y(n) > 1, 0≤ y ≤0, 0, elsewhere. Ha: 000, show that the a-level Likelihood Ratio Test 00} U {Y(n) < 000x = }. f(y|0) =
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- For each of the following distributions, compute the maximum likelihood estimator based on 7 i.i.d. observations X₁,..., X and the Fisher information, if defined. If it is not, enter DNE in each applicable input box. (a) (Enter barX_n for the sample average X Maximum likelihood estimator = X~ Ber (p), p = (0,1) Xn = 1-7 n T X₁. Hint: Use the definition of Fisher information that leads to the shorter computation. (If the Fisher information is not defined, enter DNE.) Fisher information I (p) = Use Fisher Information to find the asymptotic variance V (p) of the MLE >>. V (B) = 9 Savethe standard normal distribution is just a normal distribution with μ = 0 and σ^2 = 1. it turns out that if we standardize a normal random variable X with parameters μ, σ in the following way z = x-μ/σ the expected value of the transformation is 0 and variance is 1. so if Z = 0.2, when μ=3 and σ= 0.2, what is the normal random variable X has value?The useful life of a concrete built on the seashore has a Weibull distribution of parameter scale δ> 0 year and power β = 2. Determine a) The half life of concrete EXb) The variance VXc) The probability that the concrete lasts more than 10 years; P (X> 10)
- Suppose you have a random sample of observations from random variables Xi, i=1, 2, .., n. Assume that X;'s are identically distributed and they have the following distribution function: f(x;;0) =o* (1–ơ), x; = 1,2 , 3., 0Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.Let X1, ..,Xn be a random sample of distribution f(x; 0)=e-(=-0), x>0, 0 elsewhere, where 0 > 0. (a) Find the mle of 0. (b) For testing Họ : 0 = 00 vs H1 : 0 > 00, find the Likelihood Ratio Test (LRT) in terms of the mle found in part (a)TRUE OR FALSEa. When the assumptions of distributions are met, the parametric tests are more powerful than non-parametric testsb. In Poisson distribution, the mean is equal to its variance c. The Main objective of Statistical inference is to make inferences about the population-based n the whole set of population values d. The Variable X defined X~N(0,1) is called the Standard Normal Distribution e. Welch Test for testing two population means does not assume that the variances are equalLet X1, X2,...X, be a random sample from f(x,e) = -.0 < x< 8, then T= max(X,} is a %3D complete statistic for the parameter 0 Select one: O True FalseObtain 100 (1 x)% confidence limits (for large samples) for the parameter 2 of the f(x, 2) = e-^.^* ; x ; x = 0, 1, 2,... x! Poisson distribution:A new method has been developed in the treatment of a disease. 12 randomly selected patients were treated with this method and the time until recovery was calculated as in the picture. Establish a confidence interval for the mass mean µ. (α=0.05)Suppose X1, ..., Xn have been randomly sampled from a normal distribution with mean 0 and unknown variance sigma^2, and let U = c * i=1 -> n summation (X_ i)^2 , where c is a constant. Find the value of c that minimises the Mean Squared Error (MSE)DEx2)-Ex)(Exy) |t= (X- X)// Examus - cdn.student.uae.examus.net/?rldbqn=1&sessi... STAT-101 FEX_2021_2_Male State whether the question is true or false 4 - 30 -> 118:50 362 The degree of freedom of t-test for independent samples (where of & o are unknown and not assumed to be equal) with sample sizes n1=37 and n 43 is equal to 79. ce362d6 2dcf91 F-E <µ < i+E where E = Za/2• e362d ef917 Formulae Sheet STAT-101-2 d.f. = min (n– 1,n2 – 1) p – ERecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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