5.5.4. Let Y1, Y2, ..., Yn be a random sample from theuniform pdf fy(y; 0)=1/0, 0 ≤ y ≤ 0. Compare the Cramér-Rao lower bound for fy (y; 0) with the variance ofthe unbiased estimator ê = "+1Ymax. Discuss.

Step 1: Step 2: Step 3: Step 4:

Answered: 5.5.4. Let Y1, Y2, ..., Yn be a random…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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min{A, 2}. 6.1.13. Let X₁, X2,..., Xn be a random sample from a distribution with one of two pdfs. If 0 = 1, then f(x; 0 = 1) = √/²2T 1 -∞ < x < ∞o. If = 2, then f(x;0= 2) = 1/[(1 + x²)], -∞ < x < x. Find the mle of 0. x²/2 2п e -
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4.65 Let X1, X2, ..., Xn be a random sample from a N(u, 16) population, where u is an unknown parameter. Test Ho : u = 10 versus H1 : u+10 at a = 0.05 using the test statistic X. (a) Find the decision rule for the test. (b) Draw the power curves for n =9 and n= 100 on the same set of axes.
2.3.3. Let f(x1, 22) = 21a?r, 0 < ¤1 < c2 < 1, zero elsewhere, be the joint pdf of X1 and X2. (a) Find the conditional mean and variance of X1, given X2 = x2, 0 < 12 < 1. (b) Find the distribution of Y = E(X1 X2). (c) Determine E(Y) and Var(Y) and compare these to E(X1) and Var(X1), re- spectively.
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].
4. Let X denote the vibratory stress (psi) on a wind tur- bine blade at a particular wind speed in a wind tunnel. The article "Blade Fatigue Life Assessment with Application to VAWTS" (J. of Solar Energy Engr., 1982: 107-111) proposes the Rayleigh distribution, with pdf X f(x; 0) = { 0² e-x²/(20²) x>0 0 as a model for the X distribution. a. Verify that f(x; 0) is a legitimate pdf. b. Suppose 0 = 100 (a value suggested by a graph in the article). What is the probability that X is at most 200? Less than 200? At least 200? otherwise c. What is the probability that X is between 100 and 200 (again assuming 0 = 100)? d. Give an expression for P(X ≤ x). 5. A college professor never finishes his lecture before the

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5.5.4. Let Y1, Y2, ..., Yn be a random sample from the uniform pdf fy(y; 0) = 1/0, 0 ≤ y ≤ 0. Compare the Cramér-Rao lower bound for fy (y; 0) with the variance of the unbiased estimator ê = "+1 Ymax. Discuss.