Exponential Distribution Calculator

Calculator Inputs

Input mode

Choose whether the main input is the event rate or the mean wait.

Rate λ

Enter a positive average event rate.

x value

Used for PDF, CDF, and survival calculations.

Interval start a

Lower bound for interval probability.

Interval end b

Upper bound for interval probability.

Percentile p

Enter a decimal such as 0.90 for the 90th percentile.

Future interval s

Used in the memoryless probability calculation.

Sample size n

Used to estimate the expected total waiting time for n waits.

Graph maximum x

Sets the far right boundary of the graph.

Decimal places

Choose the displayed precision for result values.

Reset

Example Data Table

Example inputs and outputs for a quick reference scenario using λ = 0.35, x = 4, a = 2, b = 6, and p = 0.90.

Item	Value	Interpretation
Rate λ	0.350000	Average event rate per time unit.
PDF at x = 4	0.086309	Density at the selected waiting time.
CDF at x = 4	0.753403	Probability the wait is at most 4.
Survival at x = 4	0.246597	Probability the wait exceeds 4.
P(2 < X ≤ 6)	0.374129	Interval probability between the two bounds.
90th percentile	6.578815	Only 10% of waits exceed this time.
Mean	2.857143	Expected waiting time.
Variance	8.163265	Spread of waiting times.

Formula Used

Probability density function

f(x) = λe^-λx, for x ≥ 0

Cumulative distribution function

F(x) = 1 - e^-λx

Survival function

S(x) = P(X > x) = e^-λx

Interval probability

P(a < X ≤ b) = F(b) - F(a)

Quantile function

Q(p) = -ln(1-p) / λ, where 0 < p < 1

Moments and spread

Mean = 1 / λ

Variance = 1 / λ²

Standard deviation = 1 / λ

Median = ln(2) / λ

Memoryless property

P(X > x + s | X > x) = P(X > s) = e^-λs

This page also reports hazard rate, entropy, skewness, excess kurtosis, and expected total waiting time for a selected sample size.

How to Use This Calculator

Choose Rate (λ) if you know how often events occur, or Mean if you know the average waiting time.
Enter the main parameter value, then provide a nonnegative x value for pointwise probability calculations.
Enter a and b to measure the probability that the waiting time falls inside an interval.
Enter a decimal percentile such as 0.95 to get the corresponding quantile.
Use future interval s to evaluate the memoryless probability of waiting at least s more units.
Set a sample size to estimate the expected total waiting time across multiple observations.
Choose the graph maximum and decimal precision, then submit the form.
Review the results above the form, inspect the graph, and export the table as CSV or PDF.

Frequently Asked Questions

1) What does the exponential distribution model?

It models waiting times between independent events that occur at a constant average rate. Common examples include failures, arrivals, calls, and service completions.

2) When should I use it instead of a normal distribution?

Use it for nonnegative, right-skewed waiting times or gap lengths. It is not a good choice for symmetric measurements that cluster around a central value.

3) What is λ in this calculator?

λ is the event rate. A larger λ means events happen sooner, the average wait becomes shorter, and the density drops more quickly.

4) What does memoryless mean?

It means the remaining waiting time does not depend on how long you already waited. After any elapsed time, the future behaves like a fresh exponential wait.

5) Can I enter the mean instead of the rate?

Yes. Choose the mean input mode, and the calculator converts it internally using λ = 1 / mean before computing all probabilities and statistics.

6) Why must x, a, and b be nonnegative?

The exponential distribution only applies on values starting from zero. Negative inputs fall outside the model’s valid support and would produce invalid interpretations.

7) What does the interval probability represent?

It is the probability that the waiting time falls between two bounds. The calculator finds it by subtracting F(a) from F(b).

8) What does the percentile result mean?

A percentile gives the waiting time below which a chosen share of outcomes falls. The 90th percentile is exceeded by only 10% of waits.