2. Let X~ f(x; 0) for 0 € © and suppose X = {x : f(x; 0) > 0} does notdepend on 0. If2²дөдхf(x; 0). :ƒ(x; 0) >ə20əf(x; 0)ƒ(x;0)for almost every (x,0) = X × 0, then ƒ(x; 0) has monotone likelihood ratioin x.

When an input variable increases, a monotone function always either increases or always decreases.…

Answered: 2. Let X~ f(x; 0) for 0 € and suppose X…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Show that a gamma distribution with α > 1 has a rel-ative maximum at x = β(α − 1). What happens when 0 <α< 1 and when α = 1?
The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…
5. Does f(x) = In(x) satisfy the hypotheses of the Mean Value Theorem on [1,4]? If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.
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2. Let X~ f(x; 0) for 0 € and suppose X ={x : f(x; 0) > 0} does not depend on 0. If 8² әөдх ƒ(x; 0); -ƒ(x;0) > ә df f (x; 0) ƒ (x; 0) Әх' for almost every (x,0) = X × 0, then f(x; 0) has monotone likelihood ratio in x.