## Exponential Distribution

**The exponential distribution** is a continuous probability distribution that models the **time between events** in a Poisson process.

We remember that the Poisson process describes the number of events that occurred during some period. On the other hand, exponential distribution describes the **time between the occurrence of two successive events** (the distance between two adjacent non-zero points in the graph above).

Poisson distribution is parameterized by the `mu`

parameter, which describes the average number of accidents by the time unit. The exponential distribution is parametrized by the parameter `scale`

which determines the **average time between two accidents**.

Note

There is a clear relationship between the Poisson process parameter

`mu`

and the exponential distribution parameter`scale`

:`mu`

=`1 \ scale`

for one unit of time

## Example

Suppose the average time between customer arrivals at a store is `5`

minutes. What is the probability that the next customer arrives within `3`

minutes?

We have a Poisson process where an event is the customer's arrival. The average time between two arrivals is `5`

minutes. As a result, we can use exponential distribution to calculate the corresponding probability:

We also use `.cdf()`

method of the `scipy.stats.expon`

class with a specified `scale`

parameter to calculate the corresponding probability on the interval `[0,3]`

.

Everything was clear?

Course Content

Probability Theory Basics

# Probability Theory Basics

5. Covariance and Correlation

## Exponential Distribution

**The exponential distribution** is a continuous probability distribution that models the **time between events** in a Poisson process.

We remember that the Poisson process describes the number of events that occurred during some period. On the other hand, exponential distribution describes the **time between the occurrence of two successive events** (the distance between two adjacent non-zero points in the graph above).

Poisson distribution is parameterized by the `mu`

parameter, which describes the average number of accidents by the time unit. The exponential distribution is parametrized by the parameter `scale`

which determines the **average time between two accidents**.

Note

There is a clear relationship between the Poisson process parameter

`mu`

and the exponential distribution parameter`scale`

:`mu`

=`1 \ scale`

for one unit of time

## Example

Suppose the average time between customer arrivals at a store is `5`

minutes. What is the probability that the next customer arrives within `3`

minutes?

We have a Poisson process where an event is the customer's arrival. The average time between two arrivals is `5`

minutes. As a result, we can use exponential distribution to calculate the corresponding probability:

We also use `.cdf()`

method of the `scipy.stats.expon`

class with a specified `scale`

parameter to calculate the corresponding probability on the interval `[0,3]`

.

Everything was clear?