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 *λ*1 and *λ*2 are Poisson random variables, respectively. For independent X and Y, X + Y is a Poisson random variable with parameters *λ*1+*λ*2. Here is a formula that explains its distribution