Exponential Distribution Calculator
Compute probabilities, percentiles, and summary stats for an exponential random variable using rate parameter λ.
What is the exponential distribution?
The exponential distribution models waiting time until the next event in a process that happens continuously and independently at a constant average rate. Typical examples include time between incoming calls, time until a component fails, or time between arrivals in a queue.
If X ~ Exponential(λ), then the parameter λ is the event rate (events per unit time). Larger values of λ imply shorter expected waiting times.
P(X > s + t | X > s) = P(X > t).
In plain language, “how long you already waited” does not change the remaining distribution.
Core formulas used by this calculator
1) Probability density function (PDF)
f(x) = λ e-λx, for x ≥ 0.
2) Cumulative distribution function (CDF)
F(x) = P(X ≤ x) = 1 - e-λx, for x ≥ 0.
3) Survival function
S(x) = P(X > x) = e-λx.
4) Interval probability
P(a < X ≤ b) = F(b) - F(a) = e-λa - e-λb.
5) Quantile (inverse CDF)
xp = -ln(1-p)/λ, where 0 < p < 1.
How to use this exponential distribution calculator
- Enter a positive rate λ.
- Select what you want to compute: single-point metrics, interval probability, quantile, or summary stats.
- Fill in the required input fields (
x,a/b, orp). - Click Calculate to get results immediately.
If your process is described by mean waiting time instead of rate, convert first using λ = 1 / mean.
Interpretation guide
| Output | What it means |
|---|---|
| PDF at x | Relative density near x (not a direct probability at a single exact point) |
| CDF F(x) | Probability the waiting time is at most x |
| Survival S(x) | Probability the waiting time exceeds x |
| Interval probability | Probability the waiting time falls between a and b |
| Quantile xp | Time by which p proportion of events have occurred |
Worked examples
Example 1: Failure by time x
Suppose a device fails at rate λ = 0.2 per hour.
What is the probability it fails within 3 hours?
Use CDF: F(3) = 1 - e-0.2*3 = 1 - e-0.6 ≈ 0.4512.
So there is about a 45.1% chance of failure within 3 hours.
Example 2: Probability between two times
With λ = 0.5, what is P(2 < X ≤ 5)?
e-0.5*2 - e-0.5*5 = e-1 - e-2.5 ≈ 0.2858.
About 28.6%.
Example 3: 95th percentile waiting time
If λ = 1.2, find x such that F(x)=0.95.
x = -ln(1-0.95)/1.2 = -ln(0.05)/1.2 ≈ 2.496.
So 95% of waiting times are below roughly 2.50 time units.
Common mistakes to avoid
- Using a negative or zero rate parameter (λ must be strictly positive).
- Treating PDF values as direct probabilities for an exact point.
- Mixing time units (e.g., using rate per minute with x in hours).
- Using the exponential model when event rate clearly changes over time.
When exponential is appropriate (and when it is not)
The model is a strong fit when:
- Events occur independently,
- The hazard rate is approximately constant,
- You are modeling waiting time to first event.
You may need a different model when:
- Hazard increases/decreases with age (consider Weibull),
- There is a fixed minimum delay,
- Events are not independent.
Quick FAQ
Is mean equal to 1/λ?
Yes. For exponential distributions, expected waiting time is exactly 1/λ.
What is the variance?
Var(X) = 1/λ2.
Can I use this for Poisson processes?
Absolutely. The waiting time between Poisson arrivals is exponential with parameter equal to the Poisson rate.
Final thoughts
The exponential distribution is one of the most useful models in reliability, queueing, operations, and risk analysis. Use the calculator above for fast, accurate results, then interpret your outputs in practical terms: “How likely by time x?” or “How long for p% completion?” With consistent units and correct rate estimation, it becomes a powerful decision-making tool.