Binomial and Poisson distributions

The binomial distribution describes the number of successes in a fixed number of independent experiments, each with a constant probability of success. The rando...

The binomial distribution describes the number of successes in a fixed number of independent experiments, each with a constant probability of success. The random variable representing the number of successes follows a binomial distribution with parameters 'n' (number of successes) and 'p' (probability of success). The probability mass function of the binomial distribution is given by:

$P(X = k) = \left(\begin{array}{c} n \choose k \\\ p^k (1-p)^{n-k} \end{array} \right)$

where:

n! is the factorial of n
k! is the factorial of k
(n choose k) is the binomial coefficient, which is given by n! / (k! / (n-k)!)

The Poisson distribution describes the number of events occurring in a fixed interval of time or space with a constant average rate. The random variable representing the number of events follows a Poisson distribution with parameter 'λ' (average rate). The probability density function of the Poisson distribution is given by:

$f(x) = \begin{cases} \frac{\lambda^x e^{-λ}}{x!} & x = 0, 1, 2, \cdots \\\ 0 & \text{otherwise} \end{cases}$

where λ is the average rate.

The binomial and Poisson distributions are related through the following relationship:

$X \sim Bin(n, p) \quad \Leftrightarrow \quad \lambda = np$

This means that if n is fixed and p is small, the Poisson distribution and the binomial distribution are similar. As n increases, the Poisson distribution approaches the binomial distribution