The Poisson Probability Distribution

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.

Poisson Distribution
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#: \[X \sim Pois(\lambda)\]

GeoGebra

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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 x occurrences), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x occurrences), #\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.

Suppose #X# has a Poisson distribution with #\lambda=5#.

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

#\mathbb{P}(X = 6)=0.146#

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.

Excel Calculation

To calculate #\mathbb{P}(X = 6)# 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 x occurrences), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x occurrences), #\mathbb{P}(X = x)#

Thus, to calculate #\mathbb{P}(X = 6)#, we run the following command:
\[= \text{POISSON.DIST}(6, 5, \text{FALSE})\]
This gives:
\[\mathbb{P}(X = 6) = 0.146\]

R Calculation

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

New example

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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 x occurrences), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x occurrences), #\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.

Cumulative Properties of Discrete Distributions
Since the geometric distribution is discrete, the following properties regarding its cumulative probabilities hold:

#\mathbb{P}(X \leq x) = \mathbb{P}(X=0) +\mathbb{P}(X=1) + \ldots + \mathbb{P}(X=x)#
#\mathbb{P}(X \lt x) = \mathbb{P}(X \leq x-1)#
#\mathbb{P}(X \gt x) = 1 - \mathbb{P}(X \leq x)#
#\mathbb{P}(X \geq x) = 1 - \mathbb{P}(X \leq x-1)#

Additionally, for any #2# values #a# and #b# in the range of #X# (with #a<b#):

#\mathbb{P}(a \leq X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a - 1)#
#\mathbb{P}(a \leq X \lt b) = \mathbb{P}(X \leq b - 1) - \mathbb{P}(X \leq a - 1)#
#\mathbb{P}(a \lt X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a)#
#\mathbb{P}(a \lt X \lt b) = \mathbb{P}(X \leq b - 1) - \mathbb{P}(X \leq a)#

Suppose #X# has a Poisson distribution with #\lambda=4.5#.

Compute #\mathbb{P}(X \leq 1)#. Round your answer to #3# decimal places.

#\mathbb{P}(X \leq 1)=0.061#

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.

Excel Calculation

To calculate #\mathbb{P}(X \leq 1)# 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 x occurrences), #\mathbb{P}(X \leq x)#

FALSE - uses the probability mass function (exactly x occurrences), #\mathbb{P}(X = x)#

Thus, to calculate #\mathbb{P}(X \leq 1)#, we run the following command:
\[= \text{POISSON.DIST}(1, 4.5, \text{TRUE})\]
This gives:
\[\mathbb{P}(X\leq 1) = 0.061\]

R Calculation

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:
\[\text{ppois}(q = 1, lambda = 4.5)\]
This gives:
\[\mathbb{P}(X \leq 1) = 0.061\]

New example

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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:\[\sigma = \sqrt{\lambda}\]

Let #X# be a Poisson distributed random variable with parameter #\lambda=1.5#.

Calculate the expected value of #X#.

#\mu=1.5#

The expected value of a Poisson random variable #X\sim Pois(\lambda)# is calculated as follows:
\[\begin{array}{rcl}
\mu&=& \lambda \\\\
&=& 1.5
\end{array}\]

New example