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(a) Find the probability mass function of (X₁ + X₂). (b) Using the result of (a), guess what the probability mass function of X, will be and i=1 substantiate your guess by using the method of induction. In (c), (d), (e), and (f), consider the problem of testing Ho p = 0.5 versus Ha: p = 0.2 at a = 0.038406. n (c) Write down the likelihood function L(p; x), where x = (x1, x2,,xn) is the data sample realized from the random sample X = (X₁, X2,..., Xn). points. (d) Write down the expression for the Neyman-Pearson test statistic T(x) and the corresponding rejection region T as outlined in the posted notes on Neyman-Pearson Lemma.

(a) Find the probability mass function of (X₁ + X₂). (b) Using the result of (a), guess what the probability mass function of X, will be and i=1 substantiate your guess by using the method of induction. In (c), (d), (e), and (f), consider the problem of testing Ho p = 0.5 versus Ha: p = 0.2 at a = 0.038406. n (c) Write down the likelihood function L(p; x), where x = (x1, x2,,xn) is the data sample realized from the random sample X = (X₁, X2,..., Xn). points. (d) Write down the expression for the Neyman-Pearson test statistic T(x) and the corresponding rejection region T as outlined in the posted notes on Neyman-Pearson Lemma.

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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This problem explores the question of testing Ho p= 0.5 versus Ho p < 0.5, based on a random sample X₁, X2, ,, Xn from the Geometric (p) distribution. Recall that there is an ambiguity surrounding the definition of the Geometric (p) distribution, so let us unequivocally specify which distribution we are going to work with. In the experiment of repeatedly tossing a coin with the probability of getting a Head in a single toss of the coin equal to p = (0, 1) till a Head is obtained, let X denote the number of Tails obtained before the experiment is terminated. Then X has the Geometric (p) distribution. Further recall that X₁, X2,, Xn is a random sample of size n from the Geometric (p) distribution means X = (X₁, X2,, Xn) is a collection of mutually independent random variables, with each X, having the Geometric (p) distribution. Before proceeding further, let us recall the following definition. Definition A collection of random variables {Y₁, Y2,..., Yr} defined on the same sample space, with S, denoting the support of Y, for 1 ≤ i ≤r, is said to be mutually independent if, \ (81, 82, ..., sr) €, S₁, the Cartesian product of (S₁, S2, · · ·, Sr), P(₁₂(Y₁ = $i)) = [[P(Y₁ = $₁). 1=1 Remember that if {Y₁, Y2,, Yr} is a collection of mutually independent random variables, then for any 1 ≤ m ≤ r, the random variables U = g(Y₁, Y2,, Ym) and V = h(Ym+1, Ym+2,, Y₂) are independent.
Transcribed Image Text:This problem explores the question of testing Ho p= 0.5 versus Ho p < 0.5, based on a random sample X₁, X2, ,, Xn from the Geometric (p) distribution. Recall that there is an ambiguity surrounding the definition of the Geometric (p) distribution, so let us unequivocally specify which distribution we are going to work with. In the experiment of repeatedly tossing a coin with the probability of getting a Head in a single toss of the coin equal to p = (0, 1) till a Head is obtained, let X denote the number of Tails obtained before the experiment is terminated. Then X has the Geometric (p) distribution. Further recall that X₁, X2,, Xn is a random sample of size n from the Geometric (p) distribution means X = (X₁, X2,, Xn) is a collection of mutually independent random variables, with each X, having the Geometric (p) distribution. Before proceeding further, let us recall the following definition. Definition A collection of random variables {Y₁, Y2,..., Yr} defined on the same sample space, with S, denoting the support of Y, for 1 ≤ i ≤r, is said to be mutually independent if, \ (81, 82, ..., sr) €, S₁, the Cartesian product of (S₁, S2, · · ·, Sr), P(₁₂(Y₁ = $i)) = [[P(Y₁ = $₁). 1=1 Remember that if {Y₁, Y2,, Yr} is a collection of mutually independent random variables, then for any 1 ≤ m ≤ r, the random variables U = g(Y₁, Y2,, Ym) and V = h(Ym+1, Ym+2,, Y₂) are independent.
(a) Find the probability mass function of (X₁ + X₂). n (b) Using the result of (a), guess what the probability mass function of EX, will be and i=1 substantiate your guess by using the method of induction. In (c), (d), (e), and (f), consider the problem of testing at a = 0.038406. Ho : p 0.5 versus Ha: p = 0.2 (c) Write down the likelihood function L(p; x), where x = (x1, x2,,xn) is the data sample realized from the random sample X = (X1, X2,..., Xn). points. (d) Write down the expression for the Neyman-Pearson test statistic T(x) and the corresponding rejection region T as outlined in the posted notes on Neyman-Pearson Lemma.
Transcribed Image Text:(a) Find the probability mass function of (X₁ + X₂). n (b) Using the result of (a), guess what the probability mass function of EX, will be and i=1 substantiate your guess by using the method of induction. In (c), (d), (e), and (f), consider the problem of testing at a = 0.038406. Ho : p 0.5 versus Ha: p = 0.2 (c) Write down the likelihood function L(p; x), where x = (x1, x2,,xn) is the data sample realized from the random sample X = (X1, X2,..., Xn). points. (d) Write down the expression for the Neyman-Pearson test statistic T(x) and the corresponding rejection region T as outlined in the posted notes on Neyman-Pearson Lemma.
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