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 $x$. It is the value m in the probability density function $f(x)=me^{(-mx)}$ of an exponential random variable. It is also equal to $m=\frac{1}{\mu}$, where $\mu$ 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 $X \sim Exp(m)$. The mean is $\mu = \frac{1}{m}$ and the standard deviation is $\sigma = \frac{1}{m}$. The probability density function is $f(x)=me^{(-mx)}, x \geq 0$ and the cumulative distribution function is $P(X \leq x) = 1-e^{(-mx)}$.

memoryless property: For an exponential random variable $X$, 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 $X$ exceeds $x + k$, given that it has exceeded $x$, is the same as the probability that $X$ would exceed $k$ if we had no knowledge about it. In symbols we say that $P(X > x + k|X > x) = P(X > k)$.

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