The exponential distribution is a continuous probability distribution and we widely use this distribution in various fields such as engineering, research, data sciences etc.

We use this to model the time elapsed between two events in a Poisson process.

There is a very strong relationship between Poisson and exponential distribution.

For example, suppose a Poisson distribution models the number of cars passes through the highway in a given time period. We can model the time in between each car with an exponential distribution.

[For Poisson distribution- Read our article]

Here we will mathematically define the exponential distribution.

## Definition

Many problems involve the measurement of the duration of time X between an initial point of time and the occurrence of some phenomenon of interest. For example X is the life time of a light bulb, you turn it on and left until it burns out. The continuous random variable X having the probability density function

Where, the rate λ is the Normal amount of events in single time.

Here the only parameter of the distribution is λ which is greater than zero. This distribution, also known as the negative exponential distribution, is a special case of the gamma distribution (with r = 1). Examples of random variables modeled as exponential are-

a. (inter-arrival) time between two successive job arrivals

b. duration of telephone calls

c. life time (or time to failure) of a component or a product

d. service time at a server in a queue

e. time required for repair of a component

The mean value is . The median of the exponential distribution is , and the variance is shown by

Note: Both the mean and standard deviation of the exponential distribution are equal to

**Example**

**The length of time for one person to be served at a restaurant is a random variable X having an exponential distribution with a mean of 4 minutes. Find the probability that **the **restaurant s**erves a **person** **in less than 3 minutes on at least 4 of the next 6 days.**

Sol:

The probability that a a restaurant serves a person at a cafeteria in less than 3 minutes is-

Since the mean μ = 1/ λ = 4 or λ = 1/4 , the exponential distribution is-

Now

Let X represent the number of days on which a person is served in less than 3 minutes.

Then using the binomial distribution, the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days is