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3. Let = {1,2,3,4,) and C= {{1,2,3}, {4}}. Derive the sigma-algebra generated by C, i.e. that smallest sigma-algebra that contains C.

3. Let = {1,2,3,4,) and C= {{1,2,3}, {4}}. Derive the sigma-algebra generated by C, i.e. that smallest sigma-algebra that contains C.

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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II. Do what is indicated in order to answer the following problems, use only blue or black pen. Do as neatly as you can. 1. Suppose that P₁ and P2 are probabilities on (02, E) and that 0 s co s 1. Prove that P(A) = ∞ P₁(A) + ∞oP₂(A) is also a probability. 2. Five red books, six green books, and four blue books are to be arranged on a shelf. What is the probability that at least 3 green books all stand together? 3. Let = {1,2,3,4,} and C= {{1,2,3}, {4}}. Derive the sigma-algebra generated by C, i.e. that smallest sigma-algebra that contains C. 4. A person wins a contest in which he gets a free trip from San Diego (SD) to New York (NY) via Los Angeles (LA) and Chicago (CH). He has a choice of two buses from SD to LA. Once in LA, he has a choice of 3 plane flights to CH, and upon arrival in CH he has a choice of 2 trains to NY. Let & be the event that he catches bus i(i=1,2), P, the event that he catches plane /(1,2,3), and T. the event that he catches train /(-1,2). Express the following in terms of B, P, T: (a) the event that the person gets to NY; and (b) the event that the person does not get to NY. 5. Discuss the following: a) probability using axiomatic approach b) o-algebra c) mutually exclusive and independent events d) pairwise independence and independence events f) conditional probability e) subjective probability
Transcribed Image Text:II. Do what is indicated in order to answer the following problems, use only blue or black pen. Do as neatly as you can. 1. Suppose that P₁ and P2 are probabilities on (02, E) and that 0 s co s 1. Prove that P(A) = ∞ P₁(A) + ∞oP₂(A) is also a probability. 2. Five red books, six green books, and four blue books are to be arranged on a shelf. What is the probability that at least 3 green books all stand together? 3. Let = {1,2,3,4,} and C= {{1,2,3}, {4}}. Derive the sigma-algebra generated by C, i.e. that smallest sigma-algebra that contains C. 4. A person wins a contest in which he gets a free trip from San Diego (SD) to New York (NY) via Los Angeles (LA) and Chicago (CH). He has a choice of two buses from SD to LA. Once in LA, he has a choice of 3 plane flights to CH, and upon arrival in CH he has a choice of 2 trains to NY. Let & be the event that he catches bus i(i=1,2), P, the event that he catches plane /(1,2,3), and T. the event that he catches train /(-1,2). Express the following in terms of B, P, T: (a) the event that the person gets to NY; and (b) the event that the person does not get to NY. 5. Discuss the following: a) probability using axiomatic approach b) o-algebra c) mutually exclusive and independent events d) pairwise independence and independence events f) conditional probability e) subjective probability
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