1. Review of Probability Theory

2. 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

3. 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

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

5. 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

6. 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

7. 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

8. 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