Our exponential distribution calculator can help you figure out how likely it is that a certain period of time will pass between two events. Exponential distributions are widely employed in product reliability calculations or determining how long a product will survive. A brief example would be how long your car battery lasts in months. In a nutshell, it helps us estimate the duration of time when a particular event is most likely to happen. This article will provide information about the concept of the exponential distribution, its formulaexamples, and how to use it in real life

Also, learn more about the binomial and Negative Binomial Distribution with our related post.

What is an exponential probability distribution?

In probability theory and statistics, it describes the time between events happening in a process that occurs independently and at a constant average rate, in other words, the Poisson process. There are also Probability density functions and cumulative distribution functions sometimes mentioned with the Poisson process and distributions. One of its main features is that it has no memory. So, for example, it means that the chances of an hour passing before the next train arrives at the stop are the same in the morning as in the evening. Moreover, its primary trait is that we use it to simulate the behaviour of things at a constant failure rate. 

Exponential distribution example

There are many examples in real life where we can use exponential distribution, such as predicting how much the call duration would be. For example, let’s say that according to a survey, the average time a person spends talking in one call is around 15 minutes. In that case, we can use exponential distribution to find a probability if the person will speak more or less than 15 minutes. Other such examples would be:

  • In physics, we often use it to measure radioactive decay, 
  • in engineering to measure the time associated with receiving a defective part on an assembly line,
  • we can predict when an earthquake will occur,
  • a number of cars that will pass in a minute, 
  • What would be the lifespan of our electronic gadgets, and so on.

Exponential distribution formula

The fundamental formulas for exponential distribution analysis allow you to determine whether the time between two occurrences is less than or more than X, the target time interval between events:

P(x > X) = exp(-ax)

\newline P(x ≤ X) = 1 - exp(-ax)


  • a – rate parameter of the distribution, also called decay parameter tells us how often on average the events will come, and somewhere it is expressed as \ Lambda ;
  • P(x > X) is the chance of x being higher than the given value X;
  • P(x ≤ X) the chance of x being lower than given value X;
  • exp is the exponential function.

Our calculator also includes more values:

mean \;μ = \frac{1}{a},

median \;m^2=\frac{ln(2)}{a},

variance \; σ= \frac{1}{a^2} ,

standard deviation \sigma = \sqrt{(\frac{1}{a^2})}

We also have different calculators for these values, check them out.

How to use the exponential distribution calculator?

For this example, we will assume that you run a store. The average number of customers that buy the product is 20 per hour. What will the probability be that it will take 3 minutes for a customer to appear? We will take it to step by step to solve this problem.

  1. You will need to determine your base time interval; since it is most practical, we will use 1 minute for the time interval. 
  2. rate\, perimeter we express it as “per base time interval”. After this, we need to convert 20 customers per hour to 60 minutes. In other words, it is 1 customer per 3 minutes, which makes the rate perimeter a= \frac{1}{3}.
  3. The next step is to find the value of x. in our case, it is equal to 2 minutes. Our goal is to calculate the value of P(x ≤ 2) .
  4. All that is left is to input these values to our calculator or the given formula: 

P(x ≤ X) = 1 - exp(-ax) => P(x ≤ 2) = 1 - exp(-0.33 \cdot 2) = 0.48 

The result is that there is 48% chance that you will wait less than 2 minutes for the next customer. We can also find other values that we mentioned in our calculator, all according to the formula. 

Exponential distribution variance

Variance is one of the properties of an exponential distribution. The standard formula for it is σ^2 = \frac{1}{a^2}

When we want to find the variance of the exponential distribution, we will need to find the second moment of the exponential distribution, as:

E\left [ X^2\right ]=\int_{0}^{\infty }\cdot X^2\Lambda e^-\lambda x=\frac{2}{\lambda^2}

After that, to find the variance of a continuous random variable, X, we calculate it as:

Var (X) = E(X^2)- E(X)^2 =>\,

and when we substitute mean and second moment, we get:

Var(x)=\frac{1} {\Lambda^2}. That is the variance of an exponential distribution. 

Mean of exponential distribution

Definition of mean probability and statistics is that it is an average of a dataset, and we express it with a symbol μ. It is calculated using integration by parts, and the formula is \frac{1}{\Lambda}

The \Lambda sign represents the rate perimeter, defining the mean number of events in an interval. 

Negative exponential distribution

The negative exponential distribution is used commonly as a survival distribution, describing the life span of a type of hardware put in service at what may be termed time zero. As a result, it lacks the memory attribute. Even though it is almost the same as exponential distribution, we usually called negative due to the negative sign of the exponent. Therefore, we can use it to model the duration of a repair job or time of absence of employees from their job. 

The expected value of an exponential distribution

We know it as expectation, mathematical expectation, average, mean, or first moment. It is the arithmetic mean of many independent “x”. The expected value of exponential random variable x is defined as: E(x)=\frac{1}{\Lambda}. In exponential distribution, it is the same as the mean.

Moment generating function of exponential distribution

As its name suggests, we use the moment generating function (mgf) to compute the moments of a distribution. It has great practical importance, mainly because we can use it to derive moments; its derivatives at 0 are identical to the moments of a random variable. A probability distribution, such as exponential distribution, is uniquely determined by its “mtf”. We should also say that not all random variables have a moment generating function. 


What is the parameter of an exponential distribution?

The rate parameter is the most likely number of events in the interval for each curve. And when it’s an integer, it’ll be the number of possibilities with the highest probability. We define it as the reciprocal of the scale parameter and indicate how quickly decay of the exponential function occurs.

What is Lambda in exponential distribution?

\ Lambda is sometimes also called the rate perimeter, and it determines the constant average rate at which events should happen. This means it as average time or space in-between events that follow a Poisson Distributions.

What is the variance of exponential distribution?

The variance of exponential distributions is its property, calculated after finding the second moment of the exponential distribution. We express it as Var(x)=\frac{1} {\Lambda^2}.