Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » What is Poisson Distribution?

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

What is Poisson Distribution?

Module by: Lekulana Kolobe. E-mail the author

Summary: This module introduces Poisson Distribution and gives an example of its application. It also includes an exercise to engage the reader.

Poisson distribution is a "discrete probability distribution. It expresses the probability of a number of events occurring in a fixed time if these events occur with a known average rate, and are independent of the time since the last event" [Wiki]. Such events are said to be memoryless.

Most queuing systems' characteristics such as arrival and departure processes are described by a poisson distributions. Assuming that arrivals and departures are random and independent i.e. they exhibit pure-chance property; arrivals are described by a poisson random variable or poisson random distribution as shown by equation 1.1 [Wiki, Char]

The probability that there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2, ...) is

Figure 1
Equation 1.1
Equation 1.1 (clip_image001.gif)

Where

e is the base of the natural logarithm (e = 2.71828...),

k! is the factorial of k,

λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with λ = 2.5.

Exercise 1

(taken from Bajpai A.C, 1974) The average rate of telephone calls received at an exchange of 8 lines is 6 per minute. Find the probability that a caller is unable to make a connection if this is defined to occur when all lines are engaged within a minute of the time of the call.

Solution

We first need to make an assumption that the overall rate of calls is constant, then we can use equation 1.1 as follows: Since our time unit is 1 minute, then λ = 6

Figure 2
Equation 1.1
Equation 1.1 (clip_image001.gif)
Which leads to
Figure 3
After substitution
After substitution (clip_image003.gif)
The probability of not being able to make a call occurs only when there are at least 9 calls in any interval of a minute.
Figure 4
Summation Equation
Summation Equation (clip_image005.gif)
So, this leds to p(k+1) equals to:
Figure 5
From Summation equation we get
From Summation equation we get (clip_image007.gif)
Solving this bit by bit with k from 0 to 8, we get p(k+1) = 0.8546.

References:

Extracted from " http://en.wikipedia.org/wiki/Poisson_distribution" Last Accessed on 13 March 2006

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks