Answered: For a Poisson distribution X, suppose I…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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For a Poisson distribution X, suppose I know the variance of X is 5.
(1) By property of Poisson, what is mean of X.
(2) Compute P(X≥3)
(3) Does Poisson have the property of lack of memory? List at least one name of distribution with such property.

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Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

Given information:

A random variable X following Poisson distribution with variance 5.

Therefore, $X ~ P (λ)$ and $V (X) = 5$ ------(1)

However, variance of the random variable following the Poisson distribution is:

$V (X) = λ$ -----(2)

Thus, from equation (1) and (2), X is a Poisson random variable with parameter, $λ = 5$ .

$X ~ P (5)$

Step 2

(1) If X is the Poisson random variable, the mean of X is the parameter of the random variable X.

That is,

$E (X) = λ E (X) = 5$

Therefore, with this property the mean of X is equal to the parameter, $λ = 5$ .

(2) The probability mass function of the Poisson random variable is :

$P (X = x) = \frac{e^{- λ} λ^{x}}{x!}$

The probability that X is greater than 3 is computed as:

$\begin{array}{rcl} P (X \geq 3) & = & 1 - P (X < 3) \\ = & 1 - \{P (X = 0) + P (X = 1) + P (X = 2)\} \\ = & 1 - \sum_{x = 0}^{2} \frac{e^{- λ} λ^{x}}{x!} \\ = & 1 - (\frac{e^{- 5} 5^{0}}{0!} + \frac{e^{- 5} 5^{1}}{1!} + \frac{e^{- 5} 5^{2}}{2!}) \\ = & 1 - (0.0067 + 0.0337 + 0.0842) \\ = & 1 - 0.1246 \\ = & 0.8754 \end{array}$

Therefore, $P (X \geq 3) = 0.8754$ .

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