4. Let X₁,..., Xn N(μ,2). In this question, we will construct confidence interval for ². LetC~ x²(r), and as usual for any a € [0, 1] we denote x²(r) to beP(C>x²(r)) = a.(a) Suppose that μ is known. By considering the distribution ofprove thati.i.d.prove thatXiΣ(^-^)".ni=1[Σï_₁(Xi −µ²)² Σï-₁ (Xi − µ)²]X²/2(n)Xỉa/z(n)is a 100(1-a) % confidence interval for o².(b) Suppose that is unknown. By considering the distribution of2Σ(**)(X-X) (-1).5²=[Σ,(Χ. – X) Σ(Χ. – X)2]X²/2(n-1) ¹X₁-a/2 (n − 1)is a 100(1-a) % confidence interval for o².(c) In the same setting as part (b), that is, suppose that is unknown. Construct a 100(1-a)%confidence interval for o.

Given that,Let be an independent identically normally distributed variable with mean μ and variance…

Answered: 4. Let X₁,..., Xn N(μ,²). In this…

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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9.3 Let Y₁, Y₂,..., Y, denote a random sample from the uniform distribution on the interval (0, 0+1). Let 0 =Y-; and O=Y). a Show that both ₁ and ₂ are unbiased estimators of 0. b Find the efficiency of 8₁ relative to ₂. n n+1
answer With details pls. can you explain the difference between the E(Yn) E(Yi) and E(Y(n)) .
4. Suppose X₁, ..., Xn are iid from a distribution with the pdf (e-(2-0) x > 0 f(x; 0) = for some 0 € R. Find the MLE of 0. 34 35 We went PATRULLA 15 2 271 34000 3 envelar 05352 25000- 0 otherwise
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
3. Let X1,X2, ..,X, be a random sample form a normal distribution with E(X;) = µ;and Var(X;) = o². Assume s? = E(X;_ X)²/n and s3 = (n/(n – 1))s?. Questions: a. Compare between the estimators si and s?. b. Which one you preferred? What is (are) your reason(s). 4. Let X1,X2, . X, be a random sample form a uniform distribution, f(X)e = 0*(1– e)!-*; 0 < x < 1. Assume T1 = X1 and T, = (-)Ex;. %3D a. Check the unbiasedness property for T1 and T,. b. Find the MSE of T, and T,. Can you preferer one of them? Discuss your reasons. с. 5. Suppose that X, X2,. , X, are an iid sample from the distribution given below, f(X)e = (1/2)(1+ X8); -1< X <1. Show that 3x is a consistent estimator of e.
Q1:B B. Let X and Y have a bivariate normal distribution with parameters u, = 3, ly = 1,0 = 16, o, = 25, %3! and p = Suppose that P(c,
The pressure of air (in MPa) entering a compressor is measured to be X = 8.5 ± 0.2, and the pressure of the air leaving the compressor is measured to be Y = 21.2 ± 0.3. The intermediate pressure is therefore measured to be P = √XY = 13.42 . Assume that X and Y come from normal populations and are unbiased. a) From what distributions is it appropriate to simulate values X* and Y*? b) Generate simulated samples of values X*, Y*, and P*. c) Use the simulated sample to estimate the standard deviation of P. d) Construct a normal probability plot for the values P*. Is it reasonable to assume that P is approximately normally distributed? e) If appropriate, use the normal curve to find a 95% confidence interval for the intermediate pressure.
Suppose we wish to estimate the expectation of a random variable X with a large popula- tion. We observe n members of the population, i.e. {x;}1. Individual i is included in the sample with a probability proportional to p;. Please demonstrate that the usual sample E; ®i/Pi i=1• mean ī = n-1E, xi is no longer a consistent estimator for E[X], but ã is. E; 1/Pj vi=1
Please do the following questions with full handwritten working out
Data for gas mileage (in mpg) for different vehicles was entered into a software package and part of the ANOVA table is shown below: Source DF SS MS Vehicle 2 440 220.00 Error 17 318 18.71 Total 19 758 Determine the p-value for the data. a) 0.0052 b) 0.8994 c) 0.0006 d) 0.0021 e) 0.0236 f) None of the above
5.2.1 The conditional probability distribution of Y given X = x is fy.0) = xe-y for y> 0, and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10. a. Graph fyv) = xe¬w for y > 0 for several values of x. Determine: b. P(Y < 2|X = 2) d. E(Y |X = x) f. fr(y) c. E(Y | X = 2) e. fxy(x, y)

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