A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Question
**Exercise 3.18.** Let \( X \) be a normal random variable with mean 3 and variance 4. (a) Find the probability \( P(2 < X < 6) \). (b) Find the value \( c \) such that \( P(X > c) = 0.33 \).
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Transcribed Image Text:**Exercise 3.18.** Let \( X \) be a normal random variable with mean 3 and variance 4. (a) Find the probability \( P(2 < X < 6) \). (b) Find the value \( c \) such that \( P(X > c) = 0.33 \).
# Appendix E: Table of Values for Φ(x) ## Overview The table displayed represents the cumulative distribution function (CDF) of the standard normal random variable \( Z \), defined as \( \Phi(x) = P(Z \leq x) \). This function is essential for statistical analyses involving normal distributions, as it provides the probability that \( Z \) is less than or equal to a given value \( x \). ## Table Structure The table is organized to display probabilities \( \Phi(x) \) for different values of \( x \). It is arranged with two main components: - **Row Headers**: These represent the integral part of \( x \), starting from 0.0 through 3.4 in increments of 0.1. - **Column Headers**: These represent decimal increments, ranging from 0.00 through 0.09. ### Example Interpretation - To find \( \Phi(0.53) \), locate the row for 0.5 and the column for 0.03. The intersection gives a value of \( 0.7019 \). ## Application This table is typically used to find the probability of a value falling within a given range of a normal distribution. By referencing the table, one can easily determine cumulative probabilities needed for various statistical applications.
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Transcribed Image Text:# Appendix E: Table of Values for Φ(x) ## Overview The table displayed represents the cumulative distribution function (CDF) of the standard normal random variable \( Z \), defined as \( \Phi(x) = P(Z \leq x) \). This function is essential for statistical analyses involving normal distributions, as it provides the probability that \( Z \) is less than or equal to a given value \( x \). ## Table Structure The table is organized to display probabilities \( \Phi(x) \) for different values of \( x \). It is arranged with two main components: - **Row Headers**: These represent the integral part of \( x \), starting from 0.0 through 3.4 in increments of 0.1. - **Column Headers**: These represent decimal increments, ranging from 0.00 through 0.09. ### Example Interpretation - To find \( \Phi(0.53) \), locate the row for 0.5 and the column for 0.03. The intersection gives a value of \( 0.7019 \). ## Application This table is typically used to find the probability of a value falling within a given range of a normal distribution. By referencing the table, one can easily determine cumulative probabilities needed for various statistical applications.
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