The Poisson Probability Distribution is discrete and deals with events that can only take on specific, whole number values (like the number of cars passing a certain point in an hour). Poisson Probability Distribution models the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence ($\mu$). The events must be independent of each other and occur randomly.
Table of Contents
The Poisson probability function gives the probability for the number of events that occur in a given interval (often a period of time) assuming that events occur at a constant rate during the interval.
Poisson Random Variable
The Poisson random variable satisfies the following conditions:
- The number of successes in two disjoint time intervals is independent
- The probability of success during a small time interval is proportional to the entire length of the time interval.
- The probability of two or more events occurring in a very short interval is negligible.
Apart from disjoint time intervals, the Poisson random variable is also applied to disjoint regions of space.
Applications of Poisson Probability Distribution
The following are a few of the applications of Poisson Probability Distribution:
- The number of deaths by horse kicking in the Prussian Army (it was the first application).
- Birth defects and genetic mutations.
- Rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent), especially in legal cases.
- Car accidents
- Traffic flow and ideal gap distance
- Hairs found in McDonald’s hamburgers
- Spread of an endangered animal in Africa
- Failure of a machine in one month
The formula of Poisson Distribution
The probability distribution of a Poisson random variable $X$ representing the number of successes occurring in a given time interval or specified region of space is given by
\begin{align*}
P(X=x)&=\frac{e^{-\mu}\mu^x}{x!}\,\,\quad x=0,1,2,\cdots
\end{align*}
where $P(X=x)$ is the probability of $x$ events occurring, $e$ is the base of the natural logarithm (~2.71828), $\mu$ is the mean number of successes in the given time interval (or region of space), $x$ is the number of events we are interested in, and $x!$ is the factorial of $x$.
Mean and Variance of Poisson Distribution
If $\mu$ is the average number of successes occurring in a given time interval (or region) in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to $\mu$. That is,
\begin{align*}
E(X) &= \mu\\
V(X) &= \sigma^2 =\mu
\end{align*}
A Poisson distribution has only one parameter, $\mu$ is needed to determine the probability of an event. For binomial experiments involving rare events (small $p$) and large values of $n$, the distribution of $X=$ the number of success out of $n$ trials is binomial, but it is also well approximated by the Poisson distribution with mean $\mu=np$.
When to Use Poisson Probability Distribution
The Poisson distribution is useful in various scenarios:
- Modeling Rare Events: Like accidents, natural disasters, or equipment failures.
- Counting Events in a Fixed Interval: Such as the number of customers arriving at a store in an hour, or the number of calls to a call center in a minute.
- Approximating the Binomial Distribution: When the number of trials ($n$) is large and the probability of success ($p$) is small.
It is important to note that
- The Poisson distribution is related to the exponential distribution, which models the time between events.
- It is a fundamental tool in probability theory and statistics, with applications in fields like operations research, queuing theory, and reliability engineering.
R and Data Analysis, Test Preparation MCQs
Frequently Asked Questions about Poisson Distribution
- What is the Poisson Random Variable?
- What is Poisson Probability Distribution?
- Write the Formula of Poisson Probability Distribution.
- Poisson distribution is related to what distribution?
- Give some important applications of Poisson Distribution.
- Describe the general situations in which Poisson distribution can be used.
- Name the distribution that has equal mean and variance.
- What are the required conditions for poison random variables?
