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统计与概率

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Exponential Distribution Calculator.

Calculate this statistic instantly with validated formulas.

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输入数值

输入时结果会更新。

结果: PDF f(x): 0.18393972 CDF P(X≤x): 0.63212056 Mean: 2.00000000

结果

PDF f(x): 0.18393972 CDF P(X≤x): 0.63212056 Mean: 2.00000000

Use Cases

Reliability and Failure Analysis

Use the exponential distribution to model the time until failure of components or systems, helping to estimate expected lifetimes and maintenance schedules.

Example: Enter times between failures (e.g., 100, 150, 200 hours) to estimate the failure rate.

Queueing and Waiting Time Analysis

Model the time between arrivals in a queue or the time spent waiting for service, useful in operations research and customer service planning.

Example: Input inter-arrival times (e.g., 2, 3, 5 minutes) to find the average arrival rate.

Frequently Asked Questions

What is the exponential distribution used for?
The exponential distribution models the time between events in a Poisson process, such as waiting times or equipment failure times. It is characterized by a constant rate parameter λ, and its mean is 1/λ.
How do I calculate the rate parameter (λ) from my data?
The rate parameter λ is the reciprocal of the mean of your data. First, compute the average of your values, then take 1 divided by that average. This calculator does that automatically when you enter your data.
What do the mean and variance represent in exponential distribution?
The mean is the average time between events, equal to 1/λ. The variance measures the spread of the distribution and is equal to 1/λ². Both are derived from the rate parameter.

Tips & Common Mistakes

Tips

  • Ensure your data values are positive numbers, as the exponential distribution is defined for non-negative values only.
  • Separate your values with commas or spaces; the calculator will parse them automatically.
  • The more data points you provide, the more accurate your estimate of the rate parameter will be.
  • Remember that the mean of the exponential distribution is the reciprocal of the rate parameter, so a higher rate means shorter average times.

Common Mistakes to Avoid

  • Including zero or negative values, which are not valid for exponential distribution calculations.
  • Confusing the rate parameter (λ) with the mean; they are reciprocals of each other.
  • Using data from a non-exponential process, which may lead to misleading results.

Last updated: August 13, 2026