Review of Probability Theory

Discrete uniform distribution a finite number of equally spaced values are equally likely to be observed; every one of n values has equal probability 1/n: mean , variance susceptible-infective-recovered (SIR) model of Kermack and McKendrick detailed features are not predicted by the simple SIR model, e.g. jagged features -> replacing the differential equations with equations that include stochastic processes On a longer timescale, complex pattern including extinctions and re-emergences missing key biological fact: there is a reservoir of plague in rodents, so it can persist for years, unnoticed by humans, and then re-emerge suddenly and explosively. -> including the rodents and spatial spread in a mathematical model Probability mass function Cumulative distribution function

Binomial distribution the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p: probability of getting exactly k successes in n trials: mean np, variance np(1 − p) Probability mass function Cumulative distribution function

Binomial distribution approximation to B(n, p) is given by the normal distribution

Discrete Poisson distribution the probability that there are exactly k occurrences, given the expected number of occurrences in this interval is λ: mean λ, variance λ Probability mass function Cumulative distribution function

Continuous uniform distribution for each member of the family, all intervals of the same length on the distribution's support are equally probable: mean , variance Probability density function Cumulative distribution function

Exponential distribution It describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate: mean λ−1, variance λ−2 Probability density function Cumulative distribution function 1 − e−λx λ e−λx

Normal distribution first approximation to describe real-valued random variables that tend to cluster around a single mean value: mean μ, variance σ2 Probability density function Cumulative distribution function