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 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.
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 x occurrences), #\mathbb{P}(X \leq x)#
- FALSE - uses the probability mass function (exactly x occurrences), #\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 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.
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 x occurrences), #\mathbb{P}(X \leq x)#
- FALSE - uses the probability mass function (exactly x occurrences), #\mathbb{P}(X = x)#
\[= \text{POISSON.DIST}(1, 4.5, \text{TRUE})\]
This gives:
\[\mathbb{P}(X\leq 1) = 0.061\]
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:
\[\mathbb{P}(X \leq 1) = 0.061\]
#\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:
\[\begin{array}{rcl}
\mu&=& \lambda \\\\
&=& 1.5
\end{array}\]
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