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Problem 1: Suppose that Y1, Y2, . Y is an IID sample of size n from Bernoulli(p) random variables. Recall that the Bernoulli(p) model is the same as the B(n, p) model with n = 1. That is the probability mass function of Y is P(Y = k) = p(1- p)'-* for k e (0,1}. (a) Argue, using the Law of Large Numbers that Y converge in probability to p (b) Use the central limit theorem to find the limiting distribution of n(Y-p) (c) If p is the probability of success, define the quantity 7(p) = The quantity 7(p) is called the odds of occurrence of the event if p=P(success). Use the continuous mapping theorem to show that r(Y) T(p). (d) Use the delta method to find the limiting distribution of

Problem 1: Suppose that Y1, Y2, . Y is an IID sample of size n from Bernoulli(p) random variables. Recall that the Bernoulli(p) model is the same as the B(n, p) model with n = 1. That is the probability mass function of Y is P(Y = k) = p(1- p)'-* for k e (0,1}. (a) Argue, using the Law of Large Numbers that Y converge in probability to p (b) Use the central limit theorem to find the limiting distribution of n(Y-p) (c) If p is the probability of success, define the quantity 7(p) = The quantity 7(p) is called the odds of occurrence of the event if p=P(success). Use the continuous mapping theorem to show that r(Y) T(p). (d) Use the delta method to find the limiting distribution of

MATLAB: An Introduction with Applications
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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Problem 1: Suppose that Y1, Y2, . Yn is an IID sample of size n from Bernoulli(p) random variables. Recall that the Bernoulli(p) model is the same as the B(n, p) model with n = 1. That is the probability mass function of Y is P(Y = k) = p*(1– p)'-k for k e {0,1}. (a) Argue, using the Law of Large Numbers that Y converge in probability to p (b) Use the central limit theorem to find the limiting distribution of n(Y – p) (c) If p is the probability of success, define the quantity r(p) =L. The quantity T(p) is called the odds of occurrence of the event if p=P(success). Use the continuous mapping theorem to show that 7(Y) r(p). (d) Use the delta method to find the limiting distribution of Vn ( +(p)
Transcribed Image Text:Problem 1: Suppose that Y1, Y2, . Yn is an IID sample of size n from Bernoulli(p) random variables. Recall that the Bernoulli(p) model is the same as the B(n, p) model with n = 1. That is the probability mass function of Y is P(Y = k) = p*(1– p)'-k for k e {0,1}. (a) Argue, using the Law of Large Numbers that Y converge in probability to p (b) Use the central limit theorem to find the limiting distribution of n(Y – p) (c) If p is the probability of success, define the quantity r(p) =L. The quantity T(p) is called the odds of occurrence of the event if p=P(success). Use the continuous mapping theorem to show that 7(Y) r(p). (d) Use the delta method to find the limiting distribution of Vn ( +(p)
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