5. Let P be a probability measure and A, B two events. Show that |P(An B) – P(A)P(B)| < . Hìnt: Show that the two inequalities P(AN B) – P(A)P(B) < and P(A)P(B) – P(An B) < are true. To show the first of these inequalities, use the fact that the function f(p) = p(1 – p), for p e [0, 1], assumes its Maximum 1/4 at the point p = 1/2, so f(p) = 1/4 and f(p) < 1/4 for all p € [0, 1]. To show the second of these inequalities, use the first inequality with Be instead of B.
5. Let P be a probability measure and A, B two events. Show that |P(An B) – P(A)P(B)| < . Hìnt: Show that the two inequalities P(AN B) – P(A)P(B) < and P(A)P(B) – P(An B) < are true. To show the first of these inequalities, use the fact that the function f(p) = p(1 – p), for p e [0, 1], assumes its Maximum 1/4 at the point p = 1/2, so f(p) = 1/4 and f(p) < 1/4 for all p € [0, 1]. To show the second of these inequalities, use the first inequality with Be instead of B.
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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![5. Let P be a probability measure and A, B two events. Show that |P(An
B) – P(A)P(B)| < 1.
Hint: Show that the two inequalities
P(AN B) – P(A)P(B) <
and
P(A)P(B) – P(An B) <
are true. To show the first of these inequalities, use the fact that the
function f(p) = p(1 – p), for p e [0, 1], assumes its Maximum 1/4 at
the point p = 1/2, so f(p) = 1/4 and f(p) < 1/4 for all p € [0, 1]. To
show the second of these inequalities, use the first inequality with Be
instead of B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd54fbc2-091c-4efc-a0ae-33f00779b90d%2F52b50741-e47d-4066-9210-4c73d3d20fb8%2Fro5rs0m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. Let P be a probability measure and A, B two events. Show that |P(An
B) – P(A)P(B)| < 1.
Hint: Show that the two inequalities
P(AN B) – P(A)P(B) <
and
P(A)P(B) – P(An B) <
are true. To show the first of these inequalities, use the fact that the
function f(p) = p(1 – p), for p e [0, 1], assumes its Maximum 1/4 at
the point p = 1/2, so f(p) = 1/4 and f(p) < 1/4 for all p € [0, 1]. To
show the second of these inequalities, use the first inequality with Be
instead of B.
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