Poisson Distribution

Poisson Distribution

Poisson Distribution is a Discrete Probability Distribution. It is associated with the outcomes of experiments which are either success or failure. They occur during a time interval or in a specified region. These experiments are also called Poisson Experiments. Poisson Experiments are derived from the Poisson Process.

Properties of Poisson Process

• The Poisson Process has no memory. This means that the number of outcomes occurring in a time interval or region are independent of the number of outcomes occurring in some other time interval.

• The probability that an outcome will occur in a small time interval or small region is proportional to the length of the time interval or the size of the region.

• The probability that more than one outcome will occur in such a short time interval or fall in such a small region is negligible.

Probability Mass Function

The Probability Mass Function for a Poisson Distribution is

$p(x) =\frac{e^{-\lambda t}(\lambda t)^x}{x!}, x = 0,1,2...$

• $e=2.71828 . . .$
• $x$ is the value associated with the outcome of the Random Variable X.
• $\lambda$ is the average number of outcomes per unit time, distance, area or volume.
• $t$ is the number of outcomes occurring in a given time interval or specified region.

Example

During a laboratory experiment, the average number of radioactive particles passing through a counter in 1 millisecond is four. What is the probability that six particles enter the counter in a given millisecond?

Solution

Here we can extract the following things out of the question

• $\lambda t=4$
• $x=6$

$p(6)=\frac{e^{-4}(4)^{6}}{6!}=\frac{75.0208}{720}=0.1042$

So, the probability that six particles enter the counter in a given millisecond is 0.1042.