In instances in which there are a large number of possible outcomes, the Poisson distribution may be applied. The properties of the Poisson distribution are outlined below.
Expected Value of Poisson Random Variable:
In a Poisson distribution with parameter λ, a discrete random variable X is expected to have a value of
E[X]=λ.
Variance of Poisson Random Variable:
The variance of a discrete random variable X following a Poisson distribution with parameter λ will be
Var[X]=λ.
Mode of Poisson Random Variable:
If A Poisson distribution with parameter λ has a mode of ⌊λ⌋. , which is not an integer.Therefore , both λ and λ−1 are modes.
Median of Poisson Random Variable:
When a Poisson distribution is characterized by parameter λ, its median ρ satisfies the following formula
λ−ln2≤ρ≤λ+ 1/3
Sum of Independent Poisson Random Variables:
For X and Y, assume the parameters λ1 and λ2 are Poisson random variables, respectively. For independent X and Y, X + Y is a Poisson random variable with parameters λ1+λ2. Here is a formula that explains its distribution
