Exponential Distribution Calculator

Enter the rate parameter λ and evaluate exponential probabilities, density, and descriptive statistics in one place.

Rate parameter (λ)

Must be positive; higher λ corresponds to shorter expected waiting times.

Evaluate at x

Set x ≥ 0 to compute PDF, CDF, and survival probability P(X > x).

Mean (1/λ)

1.667

Variance (1/λ²)

2.778

Std. Deviation

1.667

Median

1.155

Mode

0.000

PDF at x

0.1807

f(x) = λ e^−λx

CDF at x

0.6988

F(x) = 1 − e^−λx

Survival Probability

30.12%

S(x) = P(X > x) = e^−λx

Density samples (copy into spreadsheets or plotting tools)

x	f(x)
0.00	0.6000
0.17	0.5429
0.33	0.4912
0.50	0.4445
0.67	0.4022
0.83	0.3639
1.00	0.3293
1.17	0.2980
1.33	0.2696
1.50	0.2439
1.67	0.2207
1.83	0.1997
2.00	0.1807
2.17	0.1635
2.33	0.1480
2.50	0.1339
2.67	0.1211
2.83	0.1096
3.00	0.0992
3.17	0.0897
3.33	0.0812
3.50	0.0735
3.67	0.0665
3.83	0.0602
4.00	0.0544
4.17	0.0493
4.33	0.0446
4.50	0.0403
4.67	0.0365
4.83	0.0330
5.00	0.0299
5.17	0.0270
5.33	0.0245
5.50	0.0221
5.67	0.0200
5.83	0.0181
6.00	0.0164
6.17	0.0148
6.33	0.0134
6.50	0.0121
6.67	0.0110
6.83	0.0099
7.00	0.0090
7.17	0.0081
7.33	0.0074
7.50	0.0067
7.67	0.0060
7.83	0.0055
8.00	0.0049
8.17	0.0045
8.33	0.0040
8.50	0.0037
8.67	0.0033
8.83	0.0030
9.00	0.0027
9.17	0.0025
9.33	0.0022
9.50	0.0020
9.67	0.0018
9.83	0.0016
10.00	0.0015
10.17	0.0013
10.33	0.0012
10.50	0.0011
10.67	0.0010
10.83	0.0009
11.00	0.0008
11.17	0.0007
11.33	0.0007
11.50	0.0006
11.67	0.0005
11.83	0.0005
12.00	0.0004
12.17	0.0004
12.33	0.0004
12.50	0.0003
12.67	0.0003
12.83	0.0003
13.00	0.0002
13.17	0.0002
13.33	0.0002
13.50	0.0002
13.67	0.0002
13.83	0.0001
14.00	0.0001
14.17	0.0001
14.33	0.0001
14.50	0.0001
14.67	0.0001
14.83	0.0001
15.00	0.0001
15.17	0.0001
15.33	0.0001
15.50	0.0001
15.67	0.0000
15.83	0.0000
16.00	0.0000
16.17	0.0000
16.33	0.0000
16.50	0.0000
16.67	0.0000

How to Use This Calculator

Enter a positive rate λ describing how often events occur (events per unit time/space).
Optionally provide an x value to evaluate the cumulative probability and density at that point.
Review the summary cards for mean, variance, median, and survival probability.
Export the density sample table for quick plotting or further analysis.

Formula Reference

f(x) = λ e^−λx, x ≥ 0

F(x) = 1 − e^−λx

S(x) = e^−λx

Mean = 1 / λ

Variance = 1 / λ²

Median = (ln 2) / λ

The exponential distribution models waiting times between independent Poisson events. It is memoryless, meaning the probability of waiting an additional time t does not depend on how much time has already elapsed.

Frequently Asked Questions

What does λ represent?

λ is the arrival rate: the expected number of events per unit. A larger λ means events happen more frequently.

How is this distribution used?

It commonly models time-to-failure, service times, or the spacing of arrivals when events occur independently at a constant rate.

How do I estimate λ from data?

The maximum likelihood estimate is λ̂ = 1 / mean(data). Use a sample average of observed waiting times.

Does the exponential distribution have memory?

No. It is memoryless: P(X > s + t | X > s) = P(X > t).