Section 50 Poisson Distribution
Poisson distribution is named after the French mathematician Siméon Denis Poisson.
The Poisson distribution is designed for counts (discrete data)
The Poisson distribution is a discrete probability distribution that expresses the probability of an event occurs in a fixed interval of time or space given the event occurs with a known constant rate and independently of the time since the last event.
The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
The Binomial distribution is appropriate when we count the number of occurrences that happen out of a total number possible. - number of patients with a specific disease out of a group of N individuals - number of seeds that germinate out of a total sown
The Poisson distribution is often used to describe count data, where events are occurring randomly in time or space.
The Poisson distribution can take any positive integer, including zero.
The Poisson distribution assumes that counts are obtained for n independent samples, all taken from an underlying population in which the mean count is \(\lambda\)
It is characterized by a single parameter, its mean, sometimes denoted by the Greek letter ‘lambda’: \(\lambda\).
The parameter \(\lambda\) is the average number of occurrences per unit time or space, for example, the average number of bacterial colony per square cm on a plate, the average number of cars in a queue, the average number of eggs in a nest, etc.
If the random variable \(X\) follows Poisson distribution with parameter \(\lambda>0\), then the probability mass function of \(X\) is:
\[ \large f(x; \lambda) = e^{-\lambda} \frac{\lambda^x}{x!} \]
We say that \(X\) has a Poisson distribution, and denote this by writing: \(X \sim Po(\lambda)\) where \(\lambda\) is known as a parameter of the distribution.
If \(X\) is a Poisson random variable with parameters \(\lambda\):
\[ \large \mu = E(X) = \lambda \]
\[ \large \sigma^2 = V(X) = \lambda \]