## Exponential Distribution

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# Exponential Distribution

Exponential distribution is mainly used to model arrival times.It simply tells us the amount of time we are required to stay put before a certain event that we are interested in occurs. It is exponential distribution that will help you answer questions such as how long you must wait in queue before you are served at a Mac Donald fast food joint.

*Definition*

Let X be a continuous random variable.Additionally, let X and *λ* be not only integers but positive ones. If the probability density function of X is given as below then X is said to have an exponential distribution with parameter *λ*.

As such, the random variable X is usually referred to as an exponential random variable. On the other hand, the parameter *λ* is often referred to as the **rate parameter.**

*Properties*

The following are the properties of any exponential random variable X:

- Mean =
*1/λ*

- Variance =
*1/**λ*^{2}

*Example*

Suppose that the average time a cashier takes to checkout a customer at a store is 4 minutes. What is the probability that the same cashier will checkout a customer in less than 2 minutes?

*Answer*

The average checkout time is 4 minutes. This will represent the mean waiting time.

i.e. µ = 4 minutes

Thus, the rate parameter is equal to 1 divided by the average checkout time.

i.e. *λ* =*1/µ* = *1/4*

Thus P(X<2) = *1-**e*^{-(1/4)(2)} = 0.3935

Hence the answer is 39.35% of the time.