The Poisson distribution is a probability distribution that describes the number of occurrences of a rare event in a fixed interval of time or space, given that the events occur independently and at a constant average rate. The distribution is named after French mathematician Siméon Denis Poisson, who first introduced it in 1837.
The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average rate of occurrences per unit of time or space. The probability mass function of the Poisson distribution is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
where X is the random variable representing the number of occurrences
k is a non-negative integer
e is the mathematical constant approximately equal to 2.71828
k! is the factorial of k.
The Poisson distribution has several important properties:
- Mean: The mean of the Poisson distribution is equal to λ.
- Variance: The variance of the Poisson distribution is also equal to λ.
- Shape: The Poisson distribution is a unimodal and right-skewed distribution, with the mode at the value of λ rounded to the nearest integer.
The Poisson distribution has a wide range of applications in various fields, including:
- Count data analysis: The Poisson distribution is often used to model count data, such as the number of accidents, defects, or arrivals in a given period of time.
- Quality control: The Poisson distribution is used to control the quality of manufactured products by monitoring the number of defects in a sample.
- Risk management: The Poisson distribution is used to model rare events, such as natural disasters or accidents, and to estimate the probability of their occurrence.
- Epidemiology: The Poisson distribution is used to model the spread of infectious diseases and to estimate the incidence and prevalence rates.
Probability function (including Poisson approximation to binomial distribution)
A probability function is a mathematical function that describes the probability of each possible outcome of a random variable. There are many different types of probability functions, including the probability mass function (PMF) and probability density function (PDF).
The PMF is used for discrete random variables and gives the probability that a random variable X takes on a specific value x:
P(X=x)
The PMF must satisfy two conditions: 1) The sum of the probabilities over all possible values of X must equal 1, and 2) The probability of any particular value of X must be between 0 and 1.
The Poisson distribution is an example of a probability function that can be used to approximate the binomial distribution when the number of trials is large and the probability of success is small. The Poisson distribution has a single parameter λ, which represents the expected number of occurrences in a fixed interval. The PMF of the Poisson distribution is given by:
P(X=k) = (e^(-λ) * λ^k) / k!
where X is the random variable representing the number of occurrences, k is a non-negative integer, e is the mathematical constant approximately equal to 2.71828, and k! is the factorial of k.
The Poisson distribution is a good approximation to the binomial distribution when the number of trials is large (n > 20) and the probability of success is small (p < 0.05), and the expected number of successes is less than 5 (λ=np < 5). In this case, the Poisson distribution can be used to estimate the probability of a specific number of successes, rather than calculating the entire binomial distribution.
The Poisson approximation to the binomial distribution is often used in practice for applications such as quality control, reliability analysis, and insurance risk modeling.
Constants
The Poisson distribution is characterized by a single parameter λ (lambda), which represents the average rate of occurrences per unit of time or space. The parameter λ is also equal to the mean and variance of the Poisson distribution.
Some common constants associated with the Poisson distribution include:
Probability mass function (PMF): The PMF of the Poisson distribution is given by the formula:
P(k) = (e^(-λ) * λ^k) / k!
where k is the number of occurrences and λ is the parameter of the Poisson distribution. This formula provides the probability of observing k occurrences in a given time or space interval, assuming that the occurrences are independent and randomly distributed.
Cumulative distribution function (CDF): The CDF of the Poisson distribution is the probability of observing k or fewer occurrences in a given time or space interval, and is given by the formula:
F(k) = ∑(i=0 to k) (e^(-λ) * λ^i) / i!
where ∑(i=0 to k) represents the sum over all values of i from 0 to k.
Expected value: The expected value of the Poisson distribution is equal to the parameter λ, and represents the average number of occurrences in a given time or space interval.
Variance: The variance of the Poisson distribution is also equal to the parameter λ, and represents the spread or variability of the number of occurrences in a given time or space interval.
Moment generating function: The moment generating function of the Poisson distribution is given by the formula:
M(t) = e^(λ(e^t – 1))
where t is a variable and λ is the parameter of the Poisson distribution. The moment generating function provides a way to compute moments of the distribution, such as the mean and variance.
Fitting of Poisson distribution
The Poisson distribution is characterized by a single parameter λ (lambda), which represents the average rate of occurrences per unit of time or space. The parameter λ is also equal to the mean and variance of the Poisson distribution.
To fit a Poisson distribution to a set of data, we need to estimate the parameter λ from the data. The maximum likelihood estimation (MLE) method is commonly used for this purpose. The MLE method finds the value of λ that maximizes the likelihood function of the data, which is the probability of observing the data given a specific value of λ.
The likelihood function of the Poisson distribution is given by:
L(λ) = ∏(e^(-λ) * λ^k) / k!
Where
∏ represents the product over all the observed values of the random variable, k.
To find the maximum likelihood estimate of λ, we need to maximize the likelihood function with respect to λ. Taking the natural logarithm of the likelihood function and setting its derivative with respect to λ equal to zero yields the following equation:
d/dλ ln(L(λ)) = ∑(k – λ) = 0
Solving for λ gives the maximum likelihood estimate of λ:
λ = ∑k / n
where n is the sample size and ∑k is the sum of the observed values of the random variable.
Once we have estimated the parameter λ, we can use the Poisson distribution to calculate the probability of observing a specific number of occurrences or to make predictions about future occurrences based on the observed data.