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What is Exponential distribution?

A continuous probability distribution followed by the intervals between randomly occurring events and by the time to failure of a part with a constant failure rate. With rate λ, the probability that the event does not occur by time t is e^(−λt), and the mean is 1/λ. It is memoryless: the distribution of the remaining time does not depend on how much time has already passed. It is used as the model of the intervals between arrivals when the number of arrivals follows a Poisson distribution and of the time to failure in the random-failure period, and it is an assumption of the M/M/1 queueing model.

Japanese name: 指数分布

Exponential distribution appears in 3 practice questions across 2 topics, including Probability, Statistics and Queueing, Reliability.

Field: Probability, Statistics and Queueing

Try it: questions about Exponential distribution

The time to failure of a piece of equipment follows an exponential distribution with a mean of 1,000 hours. Take e^(−0.5) ≈ 0.61, e^(−1) ≈ 0.37 and e^(−1.5) ≈ 0.22. Given that the equipment has run for 500 hours without failure, which of the following is closest to the probability that it runs at least another 1,000 hours without failure?

  1. A0.22
  2. B0.37
  3. C0.61
Show answer and explanation

Answer: B. 0.37

The exponential distribution is memoryless: the distribution of the remaining time does not depend on how much time has already passed. The probability of running at least another 1,000 hours after 500 hours is therefore the same as the probability that new equipment runs at least 1,000 hours, e^(−1) ≈ 0.37. Calculating it as a conditional probability also gives P(T > 1,500) ÷ P(T > 500) = e^(−1.5) ÷ e^(−0.5) = e^(−1). Parts in the random-failure period, where the failure rate is constant, therefore do not become more likely to fail the longer they have been used.

Exponential distribution in the explanations

Questions that mention Exponential distribution (3)

  1. The time to failure of a piece of equipment follows an exponential distribution with a mean of 1,000 hours. Take e^(−0.5) ≈ 0.61, e^(−1) ≈ 0.37 and e^(−1.5) ≈ 0.22. Given that the equipment has run for 500 hours without failure, which of the following is closest to the probability that it runs at least another 1,000 hours without failure?
  2. When the failure rate λ is constant, which expression gives the probability of no failure up to time t (reliability) R(t)?
  3. Requests reach a server as Poisson arrivals (independent, random arrivals) at an average of 6 per minute. Which of the following correctly combines the distribution followed by the intervals between arrivals and their mean?

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Related terms

Conditional probability MTTF (mean time to failure) Queueing theory Bayes' theorem Binomial distribution Confidence interval Correlation coefficient Expected value Hypothesis testing Median

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