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Mathematics · Glossary

What is Exponential distribution?

Definition 35.5 High School Mathematics · Chapter 35 — Continuous Random Variables

For λ>0\lambda > 0, XX follows the exponential distribution E(λ)\mathcal E(\lambda) if its density on [0, +∞)\intco{0}{+\infty} is

f(t)=λ e−λt.f(t) = \lambda\,\eu^{-\lambda t}.

Then P(X≤x)=1−e−λx\P(X \leq x) = 1 - \eu^{-\lambda x} and P(X>x)=e−λx\P(X > x) = \eu^{-\lambda x} for x≥0x \geq 0.

The exponential density - x (here = 1): the tail area beyond t (red) is - t, and memorylessness says every tail looks like the whole distribution rescaled.
The exponential density λe−λx\lambda\eu^{-\lambda x} (here λ=1\lambda = 1): the tail area beyond tt (red) is e−λt\eu^{-\lambda t}, and memorylessness says every tail looks like the whole distribution rescaled.
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