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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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5. Suppose we have a sample space S and have defined a probability measure P. Given an event E with P(E)+ 0, we can induce a new probability measure PE over S, defined as follows: pE (A) = P(A| E) P(An E) P(E) for all ACS In class we have shown that PE satisfies the probability axiom. We therefore can use it as a regular probability measure and all the formulae for regular probability measure also hold for PE. In this assignment, you are going to verify some of such formulae explicitly. Specifically, you need to express the following formulae in terms of the original probability P and prove them explicitely in terms of Р. (a) Show that PE(A) = 1 – pE(A°) for any event A (b) Show that PE (AUB) = PE (A)+ PE (B) – PE (An B) for any two events A and B (c) If AC B, show that PE(A) < ÞE (B) (d) Give the conditional probability PE (A | B) in terms of P (e) Show that PE (AnB) = PE (A)pE (B | A) (f) If E1, E2, ..., is a partition of S, show that, for any event A, pE (A) = pE(A | E:) Þ® (E.) i=1
Transcribed Image Text:5. Suppose we have a sample space S and have defined a probability measure P. Given an event E with P(E)+ 0, we can induce a new probability measure PE over S, defined as follows: pE (A) = P(A| E) P(An E) P(E) for all ACS In class we have shown that PE satisfies the probability axiom. We therefore can use it as a regular probability measure and all the formulae for regular probability measure also hold for PE. In this assignment, you are going to verify some of such formulae explicitly. Specifically, you need to express the following formulae in terms of the original probability P and prove them explicitely in terms of Р. (a) Show that PE(A) = 1 – pE(A°) for any event A (b) Show that PE (AUB) = PE (A)+ PE (B) – PE (An B) for any two events A and B (c) If AC B, show that PE(A) < ÞE (B) (d) Give the conditional probability PE (A | B) in terms of P (e) Show that PE (AnB) = PE (A)pE (B | A) (f) If E1, E2, ..., is a partition of S, show that, for any event A, pE (A) = pE(A | E:) Þ® (E.) i=1
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