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Exponential distribution

The waiting time until the next event in a Poisson process; a memoryless lifetime.

Parameters and support

$\lambda$

rate - events per unit time - $\lambda > 0$

Support

$x \ge 0$

The formulas

f(x)

$$f(x) = \lambda e^{-\lambda x}$$

F(x)

$$F(x) = 1 - e^{-\lambda x}$$

The survival function S(x) = e^(−λx) is the form worth memorising - most questions ask for a tail.

Mean

$$E[X] = \dfrac{1}{\lambda}$$

Variance

$$\operatorname{Var}(X) = \dfrac{1}{\lambda^{2}}$$

MGF

$$M(t) = \dfrac{\lambda}{\lambda - t}, \quad t < \lambda$$

Where the moments come from

1. $S(x) = P(X > x) = e^{-\lambda x}$, so $F(x) = 1 - e^{-\lambda x}$ and $f(x) = \lambda e^{-\lambda x}$.
2. $E[X] = \int_0^\infty S(x)\,dx = \int_0^\infty e^{-\lambda x}\,dx = \frac{1}{\lambda}$.
3. $E[X^2] = \int_0^\infty 2x\,S(x)\,dx = \frac{2}{\lambda^2}$, so $\operatorname{Var}(X) = \frac{2}{\lambda^2} - \frac{1}{\lambda^2} = \frac{1}{\lambda^2}$.
4. Memorylessness follows straight from $e^{-\lambda(s+t)} = e^{-\lambda s}e^{-\lambda t}$.
5. The exponential is Gamma($\alpha = 1$, $\lambda$); a sum of $n$ independent Exponential($\lambda$) is Gamma($n$, $\lambda$).

Worked example

The mean, variance, CDF and moment generating function above are re-derived numerically from this distribution’s own density on every test run - summed over the support for a discrete distribution, integrated by quadrature for a continuous one - and compared with the closed forms printed here. A typo on this page fails that test run.

Traps

• Using λ where the mean θ = 1/λ belongs. Read whether the question gives a RATE or a MEAN.
• Doing the conditional probability the long way when memorylessness collapses it in one line.
• Confusing the exponential (a waiting time) with the Poisson (a count).

Related

• Poisson distribution
• Gamma distribution
• Geometric distribution
• Pareto distribution

Drill this: the Exam P question bank has original questions on this distribution, and the recall trainer builds its prompts from exactly the formulas above.