BYTETOOLS

Exponential Distribution Calculator

Calculate exponential distribution probabilities from a rate or a mean: PDF, CDF, survival, interval probability, median, variance and the inverse quantile.

0.18394
PDF f(x)
0.632121
CDF P(X ≤ x)
0.367879
Survival P(X > x)
0.3834
P(1 < X < 3)

Distribution summary

0.5
Rate λ
2
Mean (1/λ)
1.386294
Median (ln2/λ)
4
Variance (1/λ²)

Inverse quantile: 90.00% of waits are shorter than x = 4.60517, from x = −ln(1 − p) / λ.

Probability density curve

0246810

Shaded area = P(1 < X < 3) = 0.3834

Memorylessness

The exponential distribution is memoryless: P(X > s + t | X > s) = P(X > t). Having already waited 2 units, the chance of waiting another 2 is still 36.7879% — exactly the same as at time zero. That is why it models the time between independent random events such as arrivals, decays and failures with a constant hazard rate.

What is the Exponential Distribution Calculator?

The ByteTools Exponential Distribution Calculator models waiting times between random events. Give it either the rate λ or the average wait 1/λ — whichever your problem states — and it returns the density at a point, the cumulative probability, the survival probability, the chance of landing inside an interval and the inverse quantile.

  • Accepts either the rate λ or the mean 1/λ
  • PDF, CDF, survival function and interval probability
  • Median ln2/λ, variance 1/λ² and standard deviation
  • Inverse quantile x = −ln(1 − p)/λ for working backwards
  • Density curve with your interval shaded
  • The memoryless property demonstrated with your own numbers

How to use the Exponential Distribution Calculator

  1. 1

    Choose whether you are entering the rate λ or the mean wait 1/λ.

  2. 2

    Type that parameter, then the value x you want the probabilities at.

  3. 3

    Optionally set a lower and upper bound to get the probability of an interval.

  4. 4

    Set a quantile probability p to work backwards from a probability to a time.

  5. 5

    Read the PDF, CDF, survival and range probabilities, and check the shaded density curve.

About the Exponential Distribution Calculator

The ByteTools Exponential Distribution Calculator models waiting times between random events. Give it either the rate λ or the average wait 1/λ — whichever your problem states — and it returns the density at a point, the cumulative probability, the survival probability, the chance of landing inside an interval and the inverse quantile.

The exponential distribution is the continuous partner of the Poisson: if events arrive at a Poisson rate, the gaps between them are exponential. It is used for component lifetimes, time until the next customer, radioactive decay and queueing models. The plotted density curve shades the region you asked about so the number and the picture agree.

Everything is computed in your browser with plain double-precision arithmetic — no uploads, no accounts and no network requests of any kind — so the page keeps working offline and your figures never leave your own device. Copy the full set of results in one click when you are done.

Frequently asked questions

What is the exponential distribution used for?

It models the time until the next event when events occur randomly at a constant average rate. Common uses are time between customer arrivals, the lifetime of a component with a constant failure rate, and the gap between radioactive decays.

What is the difference between the rate and the mean?

They are reciprocals of each other. A rate of 0.5 events per hour is the same as a mean wait of 2 hours. Problems state one or the other, so this calculator accepts either and shows both once you have entered one.

Why is the median smaller than the mean?

The exponential distribution is skewed to the right, so a few very long waits drag the mean upward. The median is ln2/λ, which is about 69% of the mean 1/λ, meaning more than half of all waits are shorter than average.

What does memoryless mean in practice?

It means a component that has already survived ten hours is exactly as likely to survive another ten as a brand new one. That is realistic for a constant hazard rate but not for anything that wears out, which is why the Weibull distribution is preferred for ageing parts.

How is the exponential distribution related to the Poisson distribution?

They describe the same process from two angles. If the number of events in a fixed interval follows a Poisson distribution with rate λ, then the waiting time between consecutive events follows an exponential distribution with the same λ.

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