Prove part (i) of the continuity theorem by using the σ‐additivity (Axiom 3) of the probability measure.Prove part (ii) using part (i) and using the de Morgan laws. Theorem 13 (Continuity theorem for probability measures) Let (N, A, P) be a probability space.(i) Let A1 C A2 C.. be an increasing sequence of events in A. Then it holdslim P(A,) = P(U An\nEN(ii) Let A1 D A2 5be a decreasing sequence of events in A. Then it holds...().lim P(An)РAn

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Probability

Prove part (i) of the continuity theorem by using the σ‐additivity (Axiom 3) of the probability measure.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

Prove part (i) of the continuity theorem by using the σ‐additivity (Axiom 3) of the probability measure.
Prove part (ii) using part (i) and using the de Morgan laws.

Theorem 13 (Continuity theorem for probability measures) Let (N, A, P) be a probability space.
(i) Let A1 C A2 C.. be an increasing sequence of events in A. Then it holds
lim P(A,) = P(U An
\nEN
(ii) Let A1 D A2 5
be a decreasing sequence of events in A. Then it holds
...
().
lim P(An)
Р
An

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