A random variable $$X$$ is a function on a sample space. Typical random variables are the result of tossing a coin, the value on rolling a die, the number of aces in a Poker hand, of multiple birthdays in a company of $$n$$ people, number of successes in $$n$$ Bernoulli trials. The classical theory of probability was devoted mainly to the study of a gambler’s gain, which is again a random variable.

The position of a particle under diffusion, the energy, temperature etc. of physical systems are random variables, but they are defined in non-discrete sample spaces. The source of randomness in the random variable is the experiment itself, in which events $${X=x_{j}}$$ are observed according to the probability function $$P$$.

In a large number of repetitions of an experiment, what is the average value of the random variable? What is the variance of the random variable? The links to my notes and solved problems is given below.

Theorems and Notes