Probability
Exponential Distribution Calculator
Evaluate the exponential density and cumulative probability at a time x using the rate parameterization, and report the mean 1/lambda.
Worksheet
Press Calculate to see the result.
Evaluate the exponential density and cumulative probability at a time x using the rate parameterization, and report the mean 1/lambda.
What this calculator answers
An exponential waiting time is the continuous cousin of a geometric wait. If events arrive as a Poisson process, the time until the next event follows an exponential distribution. This page parameterizes that distribution by the rate lambda, not by the mean. The density is lambda times e to the minus lambda x, the cdf is 1 minus e to the minus lambda x, and the mean wait is the reciprocal of the rate. A scale parameterization would have asked you to type the mean directly. This one does not.
The page rejects a non-positive rate and a negative time. It then evaluates the rate-form density, the cdf, the survival probability, the mean 1/lambda, and the variance 1/lambda squared. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Exponential distribution in the rate parameterization uses this relationship.
f(x) = λ e^(−λx), F(x) = 1 − e^(−λx), mean = 1/λ, for x ≥ 0
- λ. The rate, in events per unit of time.
- x. The waiting time being evaluated.
- 1/λ. The mean wait, which is the reciprocal of the rate.
- e^(−λx). The probability of lasting longer than x.
A constant hazard lambda means the chance of the event in the next instant does not depend on how long you have already waited. Integrating that hazard from 0 to x produces the cdf 1 − e^(−λx). Differentiating the cdf produces the density, and integrating x against the density produces the mean 1/lambda. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
This page uses the rate parameterization. Lambda is 0.5 events per unit time, not the mean. The time is x = 2, which is at least 0. e^(−lambda × x) = e^(−1) = 0.367879.
The density is lambda × that exponential: 0.5 × 0.367879 = 0.18394. The cdf is 1 − e^(−lambda × x) = 0.6321. The survival P(X > x) is 0.3679.
The mean wait is 1 / 0.5 = 2, and the variance is the square of that mean, 4. A scale parameterization would have asked for the mean 2 directly. The number you typed, 0.5, is the rate.
| Density f(x) | 0.18394 |
|---|---|
| P(X ≤ x) | 0.6321 |
| P(X > x) | 0.3679 |
| Mean 1/lambda | 2 |
| Variance | 4 |
A second situation
A later time on the same rate
This page uses the rate parameterization. Lambda is 0.5 events per unit time, not the mean. The time is x = 6, which is at least 0. e^(−lambda × x) = e^(−3) = 0.049787.
The density is lambda × that exponential: 0.5 × 0.049787 = 0.024894. The cdf is 1 − e^(−lambda × x) = 0.9502. The survival P(X > x) is 0.0498.
The mean wait is 1 / 0.5 = 2, and the variance is the square of that mean, 4. A scale parameterization would have asked for the mean 2 directly. The number you typed, 0.5, is the rate.
| Density f(x) | 0.024894 |
|---|---|
| P(X ≤ x) | 0.9502 |
| P(X > x) | 0.0498 |
| Mean 1/lambda | 2 |
| Variance | 4 |
Moving x from 2 to 6 on a rate of 0.5 multiplies the exponent by 3 and drops both the density and the survival. The mean stays at 2 because the rate did not change. The nudge, by contrast, changes the rate and therefore the mean. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The rate moved from 0.5 to 1, which cuts the mean wait from 2 to 1. The headline result went from 0.18394 to 0.135335.
The nudged inputs
This page uses the rate parameterization. Lambda is 1 events per unit time, not the mean. The time is x = 2, which is at least 0. e^(−lambda × x) = e^(−2) = 0.135335.
The density is lambda × that exponential: 1 × 0.135335 = 0.135335. The cdf is 1 − e^(−lambda × x) = 0.8647. The survival P(X > x) is 0.1353.
The mean wait is 1 / 1 = 1, and the variance is the square of that mean, 1. A scale parameterization would have asked for the mean 1 directly. The number you typed, 1, is the rate.
| Density f(x) | 0.135335 |
|---|---|
| P(X ≤ x) | 0.8647 |
| P(X > x) | 0.1353 |
| Mean 1/lambda | 1 |
| Variance | 1 |
How to read the result
Read lambda as a rate, not as a mean. A lambda of 0.5 means one event every 2 time units on average. Read the cdf as the probability that the wait is at most x, and the survival as the probability that it runs longer. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The wait is continuous and cannot be negative.
- The hazard is constant, which is the memoryless property.
- Lambda is a rate, so a larger lambda means a shorter mean wait.
- The clock starts at 0 when the wait begins.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Typing the mean in the lambda box. A mean of 2 must be entered as a rate of 0.5 on this page.
- Allowing a negative time. The support starts at 0.
- Using 1 − lambda x as the cdf. That is a linear approximation, not the exponential cdf.
- Confusing this wait with the Poisson count of events. The Poisson page takes a count. This page takes a time.
Where a different calculator is the honest choice
Use the Poisson calculator for the number of events in a fixed interval. Use the geometric calculator when the wait is a count of discrete trials. Use the uniform calculator when every time in an interval is equally likely rather than memoryless. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Poisson Distribution Calculator, Geometric Distribution Calculator, Uniform Distribution Calculator.
Check yourself
For a rate of 0.5, what are the density and the cdf at time x = 2, and what is the mean?
The density is 0.5 × e^(−1) = 0.18394. The cdf is 1 − e^(−1) = 0.6321. The mean is 1/0.5 = 2. The parameter is the rate, not the mean.
Using the result in a sentence
Time between arrivals, lifetime of a component with a constant failure rate, and the wait for the next Poisson event are the settings this rate form describes. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
A hazard that rises or falls with age is not exponential. Equipment that wears out, and people whose risk changes with time, need a different survival model even if a single average rate is known. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Exponential Distribution Calculator commit to?
This page uses Exponential distribution in the rate parameterization: f(x) = λ e^(−λx), F(x) = 1 − e^(−λx), mean = 1/λ, for x ≥ 0. A constant hazard lambda means the chance of the event in the next instant does not depend on how long you have already waited. Integrating that hazard from 0 to x produces the cdf 1 − e^(−λx). Differentiating the cdf produces the density, and integrating x against the density produces the mean 1/lambda. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Rate lambda, Time x, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the density f(x) is 0.18394. Read lambda as a rate, not as a mean. A lambda of 0.5 means one event every 2 time units on average. Read the cdf as the probability that the wait is at most x, and the survival as the probability that it runs longer. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Typing the mean in the lambda box. A mean of 2 must be entered as a rate of 0.5 on this page. A hazard that rises or falls with age is not exponential. Equipment that wears out, and people whose risk changes with time, need a different survival model even if a single average rate is known. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the Poisson calculator for the number of events in a fixed interval. Use the geometric calculator when the wait is a count of discrete trials. Use the uniform calculator when every time in an interval is equally likely rather than memoryless. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The density is 0.5 × e^(−1) = 0.18394. The cdf is 1 − e^(−1) = 0.6321. The mean is 1/0.5 = 2. The parameter is the rate, not the mean. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.