A finite probability space is used to model a situation in which a random experiment with finitely many possible outcomes is conducted. We tossed a coin a finite number of times. For example, we toss the coin twice, the set of all possible outcomes is
Suppose that on each toss the probability of a head is p and the probability of a tail is q = 1 - p. We assume the tosses are independent, and so the probabilities of the individual elements. The subsets of fi are called events. For example, the event "The first toss is a head" is .The probability of an event by summing the probabilities of the elements in the event, i.e.,
Definition: A finite probability space consists of a sample space and a probability measure . The sample space is a nonempty finite set and the probability measure is a function that assigns to each element a number in [0,1] so that
An event $A$ is a subset of $\Omega$ and the probability of $A$ is defined by
Definition Let $(\Omega,{\mathbb P})$ be a finite probability space. A random variable is a real-valued function defined on $\Omega$.
Definition Let $X$ be a random variable defined on a finite probability space $(\Omega,{\mathbb P})$. The expectation (or expected valued of X) is defined to by
The variance of $X$ is defined by
Properties: The expectation and variance satisfies
Theorem (Jensen inequality): Let $X$ be a random variable on a finite probability space and $\phi$ a convex function. Then