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)
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Chapter1: Combinatorial Analysis
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The
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F230dd6fc-67da-4f36-8738-d9d162992241%2F41cc12c6-f518-4a07-8ba4-3aafc972690c%2Fjqvfkr_processed.png&w=3840&q=75)
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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