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Exponential, Gamma, and Beta

Exponential: time until the next event

If events arrive as a Poisson process at rate λ\lambda, the gap between consecutive events is exponential:

The exponential distribution
f(x)=λe−λx,x≥0F(x)=1−e−λxf(x) = \lambda e^{-\lambda x}, \quad x \geq 0 \qquad F(x) = 1 - e^{-\lambda x}

Waiting times, and the only continuous distribution with no memory of how long you have already waited.

E[X]=1λVar(X)=1λ2E[X] = \frac{1}{\lambda} \qquad \text{Var}(X) = \frac{1}{\lambda^2}

Mean and standard deviation are equal, which tells you immediately that waiting times are highly variable. At 3 trades per minute the average gap is 20 seconds, and the standard deviation is also 20 seconds.

The clean CDF makes it the friendliest continuous distribution to work with by hand. The probability of waiting more than xx is simply e−λxe^{-\lambda x}, and inverting gives the simulation recipe X=−1λln⁡(1−U)X = -\frac{1}{\lambda}\ln(1-U).

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