Exponential Distribution Calculator

Exponential distribution describes the time between events happening according to the Poisson distribution. It means that the events occur independently and at a constant average rate. Its key property is being memoryless. It means that, for example, the probability of an hour elapsing before the next bus comes to the bus stop is the same in the morning as it is in the evening. Also, the probability of a car engine breaking down during the next hour is the same during the first and hundredth hour of it running — we "forget" about the car's state. Geometric distribution also has this property.

If you want to learn more about the distributions mentioned above, visit our dedicated tools: the Poisson distribution calculator and the geometric distribution calculator.

Also, you can simulate the exponential probability distribution using the SMp(x) distribution calculator. While more versatile, it has more parameters, which makes it a bit more tricky to apply.

Some examples of cases in which the exponential distribution can be used include:

• Time between goals in a match;
• Time between two buses coming to a bus stop;
• Time between two consecutive customers in a grocery store;
• Time between failures of a machine; and
• Distance between two car accidents along a highway.

The main formulas used for the analysis of exponential distribution let you find the probability of time between two events being lower or higher than X, the target time period between events:

P(x > X) = exp(-ax)
P(x ≤ X) = 1 − exp(-ax)

where:

• a is the rate parameter of the distribution. It is the reciprocal of the average number of events in a time interval;
• P(x > X) is the probability of x being higher than the indicated value X;
• P(x ≤ X) is the probability of x being lower than the indicated value X; and
• exp denotes the exponential function. Go to the exponent calculator if you need a refresher.

You can also find other values, including:

• mean — it is equal to μ = 1/a
• median — it is equal to m = ln(2)/a
• variance — it is equal to σ² = 1/a²
• standard deviation — it is equal to σ = √(1/a²)

Let's assume you run a school cafeteria. The average number of clients is 15 students per hour. What is the probability that you will have to wait not more than 3 minutes for a client to appear? We will solve this problem step by step.

1. Determine your base time interval. In this case, it will be most practical to use 1 minute as a time interval.

2. Make sure that your rate parameter is expressed as 'per base time interval'. We need to convert 15 students per hour to 15 students per 60 minutes or 1 student per 4 minutes. Hence, the rate parameter is a = 1/4.

3. Determine the value of x. In this case, x is equal to 3 minutes. We want to calculate the value of P(x ≤ 3).

4. Input these values into the exponential distribution formula:

   P(x ≤ X) = 1 − exp(-ax)
P(x ≤ 3) = 1 − exp(-0.25 × 3) = 0.528

5. The probability of waiting less than 3 minutes is equal to 0.528, or 52.8%. If you want, you can also calculate the mean time between clients, the median, variance, and standard deviation according to the formulas above. Make sure to check the results with the exponential distribution calculator!