Calculator Inputs
Use λ for the rate per unit time. Keep x, interval values, and graph range in the same unit.
Plotly Distribution Graph
This chart shows the PDF and CDF together. The vertical marker highlights the chosen x value.
Example Data Table
These sample values help you compare different exponential rates quickly.
| Rate λ | Mean 1/λ | Median ln(2)/λ | P(X ≤ 2) | P(X > 2) |
|---|---|---|---|---|
| 0.25 | 4.000000 | 2.772589 | 0.393469 | 0.606531 |
| 0.50 | 2.000000 | 1.386294 | 0.632121 | 0.367879 |
| 1.00 | 1.000000 | 0.693147 | 0.864665 | 0.135335 |
| 1.50 | 0.666667 | 0.462098 | 0.950213 | 0.049787 |
Formula Used
Probability density function
f(x) = λe-λx, for x ≥ 0 and λ > 0.
Cumulative distribution function
F(x) = P(X ≤ x) = 1 - e-λx.
Survival function
S(x) = P(X > x) = e-λx.
Interval probability
P(a ≤ X ≤ b) = e-λa - e-λb, when 0 ≤ a ≤ b.
Quantile function
Q(p) = -ln(1 - p) / λ, for 0 < p < 1.
Moments and descriptive measures
Mean = 1/λ
Variance = 1/λ²
Standard deviation = 1/λ
Median = ln(2)/λ
How to Use This Calculator
- Enter the rate parameter λ. This must be positive.
- Add an x value to compute the PDF, CDF, and survival probability.
- Enter interval values a and b to find the probability between two points.
- Enter a percentile probability p to calculate the matching quantile.
- Adjust graph range and decimal places if needed.
- Press Calculate Now to show results above the form, update the graph, and unlock exports.
FAQs
1. What does λ mean in this distribution?
λ is the rate parameter. A larger λ means events happen faster, so waiting times become shorter and the curve drops more sharply.
2. When should I use an exponential random variable?
Use it for modeling waiting times between independent events when the event rate stays constant, such as arrivals, failures, or service completions.
3. What is the main assumption behind this model?
The model assumes a constant event rate and memorylessness. The chance of waiting longer does not depend on how long you already waited.
4. What is the difference between PDF and CDF?
The PDF measures density at one point. The CDF gives the total probability that the random variable is less than or equal to x.
5. How do I interpret the survival value?
The survival value is P(X > x). It tells you the probability that the waiting time exceeds the chosen x value.
6. Why does the mean equal 1/λ?
For an exponential distribution, average waiting time is the reciprocal of the rate. Faster rates naturally produce smaller expected waits.
7. Can this calculator find percentiles?
Yes. Enter a probability p between 0 and 1. The calculator returns the quantile where that cumulative probability is reached.
8. Do the units matter?
Yes. If λ is measured per hour, then x, interval endpoints, and the resulting mean or percentile are all expressed in hours.