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a. Let X and Y be independent exponential random variables with parameters 0, and Oy, respectively. Find the likelihood ratio test statistic A, based on X and Y, to test Ho: 0,=20y against Ha: 0#20y.

a. Let X and Y be independent exponential random variables with parameters 0, and Oy, respectively. Find the likelihood ratio test statistic A, based on X and Y, to test Ho: 0,=20y against Ha: 0#20y.

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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Part A

a. Let X and Y be independent exponential random variables with parameters 0g and Oy, respectively. Find the likelihood ratio test statistic A, based on X and Y, to test Ho: 0=20, against Ha: 0»#20y. b. Let X1,..., X25 be a random sample from N(u1, V50) and Y1,..., X2o be a random sample from N(u2, V60). We want to test the hypothesis Но : Иі — н2 3D 13 На : И1 M2#13 Find the probability of detecting a shift (power) from µi – µ2 = 13 to µi – µ2 to accept a Type I error a = 0.05. Draw the supporting graphs to show Type I and Type II errors. 15 if we are willing c. Let X1, X2,..., Xn be a random sample from N(0, o). Use the Neyman-Pearson lemma to find the best critical region for testing Ho : o? = o? Ha : o2 = o%, with o > o3. d. Refer to question (c). Suppose we are testing Но : о? Н, : о2 = 4 16 If n = 15, find a numerical value of the best critical region of size a = 0.05. Compute the power of this test.
Transcribed Image Text:a. Let X and Y be independent exponential random variables with parameters 0g and Oy, respectively. Find the likelihood ratio test statistic A, based on X and Y, to test Ho: 0=20, against Ha: 0»#20y. b. Let X1,..., X25 be a random sample from N(u1, V50) and Y1,..., X2o be a random sample from N(u2, V60). We want to test the hypothesis Но : Иі — н2 3D 13 На : И1 M2#13 Find the probability of detecting a shift (power) from µi – µ2 = 13 to µi – µ2 to accept a Type I error a = 0.05. Draw the supporting graphs to show Type I and Type II errors. 15 if we are willing c. Let X1, X2,..., Xn be a random sample from N(0, o). Use the Neyman-Pearson lemma to find the best critical region for testing Ho : o? = o? Ha : o2 = o%, with o > o3. d. Refer to question (c). Suppose we are testing Но : о? Н, : о2 = 4 16 If n = 15, find a numerical value of the best critical region of size a = 0.05. Compute the power of this test.
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