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The probability that a standard normal random variable is less than or equal to z, where z > 0, is given by the following formula in the image provided. Use the 2-point Gaussian Quadrature on a single interval to estimate Φ(2). Calculate the absolute error given that the true value is 0.9772.(Work this out by han

The probability that a standard normal random variable is less than or equal to z, where z > 0, is given by the following formula in the image provided. Use the 2-point Gaussian Quadrature on a single interval to estimate Φ(2). Calculate the absolute error given that the true value is 0.9772.(Work this out by han

A First Course in Probability (10th Edition)
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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The probability that a standard normal random variable is less than
or equal to z, where z > 0, is given by the following formula in the image provided.

Use the 2-point Gaussian Quadrature on a single interval to estimate Φ(2). Calculate the absolute error given that the true value
is 0.9772.(Work this out by hand) 

The image shows the formula for the cumulative distribution function (CDF) of the standard normal distribution, denoted as \(\Phi(z)\). The formula is: \[ \Phi(z) = \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \int_{0}^{z} \exp\left(\frac{-x^2}{2}\right) \, dx \] Explanation: - \(\Phi(z)\): Cumulative distribution function of a standard normal distribution, evaluating the probability that a normally distributed random variable is less than or equal to \(z\). - \(\frac{1}{2}\) is a constant term. - \(\frac{1}{\sqrt{2\pi}}\) is a scaling factor derived from the standard normal distribution. - \(\int_{0}^{z} \exp\left(\frac{-x^2}{2}\right) \, dx\) is an integral from 0 to \(z\) of the function \(\exp\left(\frac{-x^2}{2}\right)\), representing the area under the probability density function for a standard normal distribution from 0 to \(z\). - \(\exp(-x^2/2)\) represents the bell-shaped curve of the normal distribution.
Transcribed Image Text:The image shows the formula for the cumulative distribution function (CDF) of the standard normal distribution, denoted as \(\Phi(z)\). The formula is: \[ \Phi(z) = \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \int_{0}^{z} \exp\left(\frac{-x^2}{2}\right) \, dx \] Explanation: - \(\Phi(z)\): Cumulative distribution function of a standard normal distribution, evaluating the probability that a normally distributed random variable is less than or equal to \(z\). - \(\frac{1}{2}\) is a constant term. - \(\frac{1}{\sqrt{2\pi}}\) is a scaling factor derived from the standard normal distribution. - \(\int_{0}^{z} \exp\left(\frac{-x^2}{2}\right) \, dx\) is an integral from 0 to \(z\) of the function \(\exp\left(\frac{-x^2}{2}\right)\), representing the area under the probability density function for a standard normal distribution from 0 to \(z\). - \(\exp(-x^2/2)\) represents the bell-shaped curve of the normal distribution.
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