(c) What is the asymptotic distribution of √n(0-0)?6.2.9. If X1, X2,..., Xn is a random sample from a distribution with pdf={f(x; 0) =303(x+0)400

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Answered: 6.2.9. If X₁, X2,..., Xn is a random…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

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Chapter1: Starting With Matlab

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1.3. Let Y₁, Y₂, ..., Yn denote a random sample of size n from a population with a uniform distribution = Y(1) = min(Y₁, Y₂, ..., Yn) as an estimator for 0. Show that ê on the interval (0, 0). Consider is a biased estimator for 8.
= 7.1.7. Let X1, X2,..., Xn denote a random sample from a distribution that is N(μ,0), 0 < 0 <∞o, where u is unknown. Let Y E(Xi-X)2/n and let L[0,8(y)] = [0-8(y)]². If we consider decision functions of the form 8(y) = by, where b does not depend upon y, show that R(0, 8) = (0²/n²) [(n² - 1)6² -2n(n-1)b+n²]. Show that b = n/(n+1) yields a minimum risk decision function of this form. Note that nY/(n+1) is not an unbiased estimator of 0. With 8(y) ny/(n+1) and 0<0<∞, determine max, R(0,8) if it exists. =
1.9.18. Find the mean and the variance of the distribution that has the cdf x <0 0
1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?
3.2-5. There are times when a shifted exponential model is appropriate. That is, let the pdf of X be 1 -(xr-8)/0 8 < x < 00. = (a) Define the cdf of X. (b) Calculate the mean and variance of X.
If var(X) = var(Y) = σ², cov(X,Y)= and 2Y-3 find the correlation between 2X+3
2
4.1. Suppose X and Y are independent random variables, which are jointly continuous. x' is equal to the distribution 1. Show carefully that the distribution of 'X + Y given X of the random variable à +Y. Hint: Consider the change of variables h(x, y) = (x,x+y). = 2. Suppose X and Y each have an exponential distribution with parameter a, find the joint p.d.f of X and X + Y. 3. In the situation of (ii) above calculate E[X | X + Y].
7.4.5. Show that the first order statistic Y₁ of a random sample of size n from the distribution having pdf f(x; 0) = e-(-0), 0 < x <∞, -∞ < 0 < ∞, zero elsewhere, is a complete sufficient statistic for 0. Find the unique function of this statistic which is the MVUE of 0. 7.4.6. Let a random sample of size n be taken from a distribution of the discrete
4-52. If f(x) =e ,- co
4.5. After fitting the regression model, y= Bo + Bix1 + Brx2+ B3x3+€ on 15 cases, it is found that the mean square error s? =3 and 0.5 0.3 0.2 0.6 0.3 6.0 0.5 0.4 (X'X)-' = 0.2 0.5 0.2 0.7 0.6 0.4 0.7 3.0 Find a. The estimate of V (ß1). b. The estimate of Cov(ß¡, ß3). c. The estimate of Corr(ß1, ß3). d. The estimate of V (ß¡ – B3).

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6.2.9. If X₁, X2,..., Xn is a random sample from a distribution with pdf 303 f(x; 0) (x+0)4 = -{ 0