Random variables
A random variable assigns a numerical value to the outcome of a random experiment.
Discrete random variables
A discrete random variable takes values in a finite or countable set. For example, \(X\) might be the number of infected individuals, with probabilities \(P(X=x)\).
Continuous random variables
A continuous random variable is described by a probability density \(f(x)\), with
\[P(a\le X\le b)=\int_a^b f(x)\,dx.\]Expectation and variance
For a discrete variable,
\[E[X]=\sum_x xP(X=x),\qquad \operatorname{Var}(X)=E[X^2]-E[X]^2.\]Key idea. A random variable converts uncertain outcomes into numerical quantities whose distributions can be analysed mathematically.