### Chapter 4. Probability Distributions: Discrete Probability Distributions

### The Poisson Probability Distribution

Poisson Experiment

In a **Poisson **experiment:

- We are observing the occurrence of a particular event within a
*fixed**interval*of time or space. - The average rate of occurrence #\lambda# is known and the same for each interval of equal length.
- The probability of an event occurring is independent of the time since the last event.
- The variable of interest #X# is the number of times the event occurs within the interval.

Let #X# be the number of times a particular event occurs in a Poisson experiment, then #X# is a

**Poisson random variable**with range: \[R(X)=\{0,1,\ldots\}\]

We say that #X# is **Poisson distributed **with parameter #\lambda#: \[X \sim Pois(\lambda)\]

#\text{}#

Computation of Poisson Probabilities with Statistical Software

Let #X# be a *Poisson *random variable with parameter #\lambda#.

To compute #\mathbb{P}(X=x)# in Excel, we make use of the following function:

POISSON.DIST(x, mean, cumulative)

x: The number of occurrences.mean: The mean of the distribution.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xoccurrences), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xoccurrences), #\mathbb{P}(X = x)#

To compute #\mathbb{P}(X=x)# in R, make use of the following function:

dpois(x, lambda)

x: The number of occurrences.lambda: The mean of the distribution.

*Poisson*distribution with #\lambda=5#.

Compute #\mathbb{P}(X = 6)#. Round your answer to #3# decimal places.

There are a number of different ways we can calculate #\mathbb{P}(X = 6)#. Click on one of the panels to toggle a specific solution.

To calculate #\mathbb{P}(X = 6)# in Excel, we make use of the following function:

Thus, to calculate #\mathbb{P}(X = 6)#, we run the following command:

POISSON.DIST(x, mean, cumulative)

x: The number of occurrences.mean: The mean of the distribution.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xoccurrences), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xoccurrences), #\mathbb{P}(X = x)#

\[= \text{POISSON.DIST}(6, 5, \text{FALSE})\]

This gives:

\[\mathbb{P}(X = 6) = 0.146\]

To calculate #\mathbb{P}(X = 6)# in R, we make use of the following function:

dpois(x, lambda)

x: The number of occurrences.lambda: The mean of distribution.

Thus, to calculate #\mathbb{P}(X = 6)#, we run the following command:

\[\text{dpois}(x = 6, lambda = 5)\]

This gives:

\[\mathbb{P}(X = 6) = 0.146\]

#\text{}#

Computation of Cumulative Poisson Probabilities with Statistical Software

Let #X# be a *Poisson *random variable with parameter #\lambda#.

To calculate *cumulative *probabilities for a *Poisson* distribution in Excel, we make use of the following function:

POISSON.DIST(x, mean, cumulative)

x: The number of occurrences.mean: The mean of the distribution.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xoccurrences), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xoccurrences), #\mathbb{P}(X = x)#

To calculate *cumulative *probabilities for a *Poisson* distribution in R, make use of the following function:

ppois(q, lambda)

q: The number of occurrences.lambda: The mean of distribution.

There are a number of different ways we can calculate #\mathbb{P}(X \leq 1)#. Click on one of the panels to toggle a specific solution.

To calculate #\mathbb{P}(X \leq 1)# in Excel, we make use of the following function:

Thus, to calculate #\mathbb{P}(X \leq 1)#, we run the following command:

POISSON.DIST(x, mean, cumulative)

x: The number of occurrences.mean: The mean of the distribution.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xoccurrences), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xoccurrences), #\mathbb{P}(X = x)#

\[= \text{POISSON.DIST}(1, 3.5, \text{TRUE})\]

This gives:

\[\mathbb{P}(X\leq 1) = 0.136\]

To calculate #\mathbb{P}(X \leq 1)# in R, we make use of the following function:

ppois(q, lambda)

q: The number of occurrences.lambda: The mean of distribution.

Thus, to calculate #\mathbb{P}(X \leq 1)#, we run the following command:

\[\text{ppois}(q = 1, lambda = 3.5)\]

This gives:

\[\mathbb{P}(X \leq 1) = 0.136\]

#\text{}#

Mean, Variance, and Standard Deviation of a Poisson Random Variable

Let #X# be a *Poisson *random variable with parameter #\lambda#.

Then the **expected value** of #X# calculated with the following formula: \[\mu = \lambda\]

The **variance **of #X# is calculated with the following formula:\[\sigma^2 = \lambda\]

And the **standard deviation **of #X# is calculated with the following formula:\[\sigma = \sqrt{\lambda}\]

The

*expected value*of a Poisson random variable #X\sim Pois(\lambda)# is calculated as follows:

\[\begin{array}{rcl}

\mu&=& \lambda \\\\

&=& 0.5

\end{array}\]

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