6. The Poisson Distribution is given by Pn where > 0 is the mean and En=0 Pn = .1. The Stirling approximation for the Gamma function for large z is на е-м n! T(+1) √√27 2²+1/2e-². ≈ Pn ≈ Use this approximation to show that the Poisson Distribution becomes a Gaussian Distribution when » 1 and n » 1, i.e., that e-(n-μ)²/(2μ) 2πμ
6. The Poisson Distribution is given by Pn where > 0 is the mean and En=0 Pn = .1. The Stirling approximation for the Gamma function for large z is на е-м n! T(+1) √√27 2²+1/2e-². ≈ Pn ≈ Use this approximation to show that the Poisson Distribution becomes a Gaussian Distribution when » 1 and n » 1, i.e., that e-(n-μ)²/(2μ) 2πμ
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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How to do question 6
![**Understanding the Poisson Distribution and Its Approximation to a Gaussian Distribution**
The Poisson Distribution is mathematically represented by:
\[ p_n = \frac{\mu^n e^{-\mu}}{n!} \]
where \(\mu > 0\) is the mean, and \(\sum_{n=0}^{\infty} p_n = 1\).
**Stirling's Approximation:**
For large values of \(z\), the Stirling approximation for the Gamma function is given by:
\[ \Gamma(z + 1) \approx \sqrt{2\pi} \, z^{z + \frac{1}{2}} e^{-z} \]
**Transition to Gaussian Distribution:**
Using this approximation, we can demonstrate that the Poisson Distribution approaches a Gaussian Distribution when \(\mu \gg 1\) and \(n \gg 1\). Specifically, it can be shown that:
\[ p_n \approx \frac{e^{-(n - \mu)^2 / (2\mu)}}{\sqrt{2\pi\mu}} \]
This expression indicates that as the mean \(\mu\) becomes much larger than 1, the shape of the Poisson distribution begins to resemble that of a Gaussian (or Normal) distribution. This transition highlights an important property where discrete random variables under certain conditions begin to exhibit continuous behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b65ef36-cf51-4f81-80a6-74e205c9e9b1%2F65e8a649-1cde-4ce1-bbcc-df2e6da17a28%2Fmcr8kwa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding the Poisson Distribution and Its Approximation to a Gaussian Distribution**
The Poisson Distribution is mathematically represented by:
\[ p_n = \frac{\mu^n e^{-\mu}}{n!} \]
where \(\mu > 0\) is the mean, and \(\sum_{n=0}^{\infty} p_n = 1\).
**Stirling's Approximation:**
For large values of \(z\), the Stirling approximation for the Gamma function is given by:
\[ \Gamma(z + 1) \approx \sqrt{2\pi} \, z^{z + \frac{1}{2}} e^{-z} \]
**Transition to Gaussian Distribution:**
Using this approximation, we can demonstrate that the Poisson Distribution approaches a Gaussian Distribution when \(\mu \gg 1\) and \(n \gg 1\). Specifically, it can be shown that:
\[ p_n \approx \frac{e^{-(n - \mu)^2 / (2\mu)}}{\sqrt{2\pi\mu}} \]
This expression indicates that as the mean \(\mu\) becomes much larger than 1, the shape of the Poisson distribution begins to resemble that of a Gaussian (or Normal) distribution. This transition highlights an important property where discrete random variables under certain conditions begin to exhibit continuous behavior.
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