Poisson Distribution
Learn about Poisson Distribution as knowledge of Probability Distributions is essential in the field of Data Science and it is the backbone for understanding many concepts.
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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.
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