Key Concepts

• Exponential probability distributions often follow a decay model with higher probabilities happening for small values and lower probabilities happening for larger values.
• The Poisson distribution is an example of an exponential distribution.

Glossary

decay parameter: the rate at which probabilities decay to zero for increasing values of [latex]x[/latex]. It is the value m in the probability density function [latex]f(x)=me^{(-mx)}[/latex] of an exponential random variable. It is also equal to [latex]m=\frac{1}{\mu}[/latex], where [latex]\mu[/latex] is the mean of the random variable.

exponential distribution: a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is [latex]X \sim Exp(m)[/latex]. The mean is [latex]\mu = \frac{1}{m}[/latex] and the standard deviation is [latex]\sigma = \frac{1}{m}[/latex]. The probability density function is [latex]f(x)=me^{(-mx)}, x \geq 0[/latex] and the cumulative distribution function is [latex]P(X \leq x) = 1-e^{(-mx)}[/latex].

memoryless property: For an exponential random variable [latex]X[/latex], the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. This means that the probability that [latex]X[/latex] exceeds [latex]x + k[/latex], given that it has exceeded [latex]x[/latex], is the same as the probability that [latex]X[/latex] would exceed [latex]k[/latex] if we had no knowledge about it. In symbols we say that [latex]P(X > x + k|X > x) = P(X > k)[/latex].

Poisson distribution: If there is a known average of [latex]λ[/latex] events occurring per unit time, and these events are independent of each other, then the number of events [latex]X[/latex] occurring in one unit of time has the Poisson distribution. The probability of [latex]k[/latex] events occurring in one unit time is equal to [latex]P(X=k)= \frac{\lambda^{k} e^{- \lambda}}{k!}[/latex].