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Last updated on October 6, 2025
In statistics, the exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process. It is often used to model time to failure and waiting times. In this topic, we will learn the formula for the exponential distribution and its properties.
The exponential distribution is used to model the time between events in a process where events occur continuously and independently at a constant average rate. Let’s learn the formula to calculate probabilities in an exponential distribution.
The probability density function (PDF) of an exponential distribution is given by: \([ f(x|\lambda) = \lambda e^{-\lambda x} \text{ for } x \geq 0 ] \) where \((lambda)\) is the rate parameter.
The cumulative distribution function (CDF) is given by: \([ F(x|\lambda) = 1 - e^{-\lambda x} \text{ for } x \geq 0 ]\)
The mean of an exponential distribution is \((1/\lambda)\). The variance is\( (1/\lambda^2).\)
The median is\( ((\ln(2))/\lambda)\). The mode is 0.
Students may find the exponential distribution formula challenging, but there are ways to remember it.
In real life, the exponential distribution is widely used to model various processes:
Students often make errors when applying the exponential distribution formula. Here are some mistakes and ways to avoid them.
What is the probability that a component lasts more than 5 hours if the average rate of failure is 0.2 failures per hour?
The probability is approximately 0.368.
Given the rate \((\lambda = 0.2)\), use the CDF:\( [ F(x|\lambda) = 1 - e^{-\lambda x} ] \)
For \((x = 5)\), \([ P(X > 5) = 1 - F(5|0.2) = e^{-0.2 \times 5} \approx 0.368 ]\)
Calculate the mean time until the next event for a process with a rate of 0.5 events per minute.
The mean is 2 minutes.
The mean of an exponential distribution is given by\( (1/\lambda).\) For \((\lambda = 0.5)\), Mean = \((1/0.5 = 2)\) minutes.
What is the variance of an exponential distribution with a rate of 3 events per hour?
The variance is approximately 0.111.
The variance of an exponential distribution is \((1/\lambda^2).\) For \((\lambda = 3)\), Variance =\( (1/3^2 = 1/9 \approx 0.111).\)
If the average time between calls at a call center is 4 minutes, what is the rate of the exponential distribution?
The rate is 0.25 calls per minute.
The mean time between events is \((1/\lambda)\). Given the mean is 4 minutes, \((\lambda = 1/4 = 0.25) \)calls per minute.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.