Deterministic variable

If the outcome of a variable is fixed, i.e. if a variable will always have the exact same value, we call this a deterministic variable. 

Random or stochastic variable

A random variable is a variable, which may take a range of numerical outcomes as the value is a result of a random phenomenon. Obviously the outcome is not fixed and may differ each time. A discrete random variable can take only a countable number of outcomes; a continuous random variable takes an infinite number of possible values.

An example of a continuous random variable is a measurement of a certain quantity: repeated observations will yield different outcomes. A measurement, also called 'an observable', is a random variable. 

Notation:
\(\underline{y} \) is a random variable (denoted by underlining), \( y \) is an individual outcome or realization of \(\underline{y} \).