Problem 1: Suppose that Y, 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)'-k for k E {0,1}. (a) Argue, using the Law of Large Numbers that Y converge in probability to p (b) Usc the central limit thcorem to find the limiting distribution of n(Y- p) (c) If p is the probability of success, define the quantity T(p) = . The quantity T(p) is called %3D the odds of occurrence of the event if p=P(success). Usc the continuous mapping thcorem to show that 7(Y) 4 T(p).
Problem 1: Suppose that Y, 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)'-k for k E {0,1}. (a) Argue, using the Law of Large Numbers that Y converge in probability to p (b) Usc the central limit thcorem to find the limiting distribution of n(Y- p) (c) If p is the probability of success, define the quantity T(p) = . The quantity T(p) is called %3D the odds of occurrence of the event if p=P(success). Usc the continuous mapping thcorem to show that 7(Y) 4 T(p).
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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Please do part D, thank you!
Transcribed Image Text:Problem 1: Suppose that Y, 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)!-k for k E {0, 1}.
(a) Argue, using the Law of Large Numbers that Y converge in probability to p
(b) Usc the central limit thcorem to find the limiting distribution of n(Y- p)
(c) If p is the probability of success, define the quantity T(p) = . The quantity T(p) is called
%3D
1-p
the odds of occurrence of the event if p=P(success). Usc the continuous mapping theorem to
show that 7(Y) 4 T(p).
(d) Usc the delta method to find the limiting distribution of
Vn (F(p)
1
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