# The Four Assumptions of the Poisson Distribution

The Poisson distribution is a probability distribution that is used to model the probability that a certain number of events occur during a fixed time interval.

The Poisson distribution is appropriate to use if the following four assumptions are met:

Assumption 1: The number of events can be counted.

We assume that the number of “events” that can occur during a given time interval can be counted and can take on the values of 0, 1, 2, 3, … etc.

Assumption 2: The occurrence of events are independent.

We assume that the occurrence of one event does not affect the probability that another event will occur.

Assumption 3: The average rate at which events occur can be calculated.

We assume that the average rate at which events occur during a given time interval can be calculated and that it is constant over each sub-interval.

Assumption 4: Two events cannot occur at exactly the same instant in time.

We assume that at each extremely small sub-interval exactly one event occurs or does not occur.

The following examples show various scenarios that meet the assumptions of a Poisson distribution.

### Example 1: Number of Arrivals at a Restaurant

The number of customers that arrive at a restaurant each day can be modeled using a Poisson distribution.

This scenario meets each of the assumptions of a Poisson distribution:

Assumption 1: The number of events can be counted.

The number of customers that arrive at a restaurant each day can be counted (e.g. 200 customers).

Assumption 2: The occurrence of events are independent.

The arrival of one customer does not affect the arrival of another customer.

Assumption 3: The average rate at which events occur can be calculated.

We can easily collect data on the average number of customers that enter the restaurant each day.

Assumption 4: Two events cannot occur at exactly the same instant in time.

Two customers cannot technically enter a restaurant at exactly the same moment in time.

### Example 2: Number of Network Failures per Week

The number of network failures that a tech company experiences each week can be modeled using a Poisson distribution.

This scenario meets each of the assumptions of a Poisson distribution:

Assumption 1: The number of events can be counted.

The number of network failures each week can be counted (e.g. 3 network failures).

Assumption 2: The occurrence of events are independent.

It’s assumed that the occurrence of one network failure does not affect the probability that another network failure will occur.

Assumption 3: The average rate at which events occur can be calculated.

We can easily collect data on the average number of network failures that occur each week.

Assumption 4: Two events cannot occur at exactly the same instant in time.

Two network failures cannot occur at the exact same moment in time – only one network failure can occur at once.