Stochastic Process → Definition → Poisson Process → Definition

**Definition**

For fixed rate λ>0, a counting process is called a Poisson process {*N*(*t*), *t*∈[0,∞)} if the following three conditions are hold:

**1.** *N*(0) = 0

**2.** *N*(*t*) has independent occurrences

**3.** The occurrences in the time interval *t*>0 has Poisson distribution, i.e.

It can be written in *n* or *x*. In solving the problems on probability of a Poisson processes, we use the following shape of the formula for convenience

In all the three shapes of the formula, we have,

*t* = time interval

λ = rate

*n* or *x* = number of occurrences

**SOME EXPLANATION**

**1.** *N*(0) = 0 means the number of occurrences are zero when the counting is started. For example, if we start counting the customers in a shop that follows Poisson process at 9:00 then *N*(9) = 0

**2.** *N*(*t*) = 0 has independent occurrences means the arrivals are independent. For example, the arrivals of customers in a shop do not depend on each other. They can enter the shop when they want.

**3.** The occurrences or arrivals follows the Poisson distribution.

**NOTE:** Mean = λ*t*, so when only mean is given and not the time and rate separately, then take λ*t* = mean.