1. Definition
Exponential distribution models the wait X between Poisson events, density f(x)=λe^(−λx) (x≥0), CDF F(x)=1−e^(−λx), memoryless.
2. Symbols
| Symbol | Meaning |
|---|
| λ | rate parameter |
| x | waiting time |
| F(x)=1−e^(−λx) | cumulative probability |
| E(X)=1/λ | mean wait |
3. How it works
- CDF F(x)=1−e^(−λx);
- Survival P(X>x)=e^(−λx);
- Mean E(X)=1/λ, variance 1/λ²;
- Memoryless: P(X>s+t|X>s)=P(X>t).
4. Steps
- Enter λ, e.g. 0.5;
- Enter x, e.g. 2;
- Click Calculate for P(X≤2);
- Read the survival, mean and variance.
5. Example
Example: λ=0.5, x=2. P(X≤2)=1−e^(−1)=0.6321; mean wait 1/0.5=2.
6. Pitfalls
x cannot be negative;
Memoryless means the remaining-life distribution is the same, not “old age has no effect”;
λ is events per unit time, not the mean time.