Let X have binomial distribution with parameters n and p, find E( 1 /1 + X ) = E( integral 0 to 1 Xdt) and find the limit of this expectation as n goes to 1 and p goes to 0, the limit is taken in such a way that np goes to lamda where 0 < lamda< 1. Comment

Let X have binomial distribution with parameters n and p, find E( 1 /1 + X ) = E( integral 0 to 1 Xdt) and find the limit of this expectation as n goes to 1 and p goes to 0, the limit is taken in such a way that np goes to lamda where 0 < lamda< 1. Comment

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Let X have binomial distribution with parameters n and p, find

E( 1 /1 + X ) = E( integral 0 to 1 Xdt)

and find the limit of this expectation as n goes to 1 and p goes to 0, the limit is taken in such a way that np goes to lamda where 0 < lamda< 1. Comment

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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:…