# Probability Theory on Coin Toss Space

## Finite Probability Spaces

### Definition

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

\Omega = \{HH,HT,TH,TT\}
$\Omega = \{HH,HT,TH,TT\}$

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.,

{\mathbb P}(\text{First toss is a head})= {\mathbb P}(HT)+ {\mathbb P}(HH)
${\mathbb P}(\text{First toss is a head})= {\mathbb P}(HT)+ {\mathbb P}(HH)$

### Definition

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

\mathbb P
$\mathbb P$
\displaystyle\sum_{\omega \in \Omega} {\mathbb P} (\omega)=1
$\displaystyle\sum_{\omega \in \Omega} {\mathbb P} (\omega)=1$
{\mathbb P}(A) =\displaystyle\sum_{\omega \in A} {\mathbb P} (\omega)
${\mathbb P}(A) =\displaystyle\sum_{\omega \in A} {\mathbb P} (\omega)$

## Random Variables, Distributions, and Expectations

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

{\mathbb E}[X] = \sum_{\omega\in\Omega} X_\omega {\mathbb P}(w)
${\mathbb E}[X] = \sum_{\omega\in\Omega} X_\omega {\mathbb P}(w)$

The variance of $X$ is defined by

\text{Var}[X] = {\mathbb E}[(X-{\mathbb E}[X])^2]
$\text{Var}[X] = {\mathbb E}[(X-{\mathbb E}[X])^2]$

Properties: The expectation and variance satisfies

{\mathbb E}[c_1 X+ c_2Y] = c_1{\mathbb E}[X]+ c_2{\mathbb E}[Y]
${\mathbb E}[c_1 X+ c_2Y] = c_1{\mathbb E}[X]+ c_2{\mathbb E}[Y]$
\text{Var}[ X] = {\mathbb E}[X^2]- {\mathbb E}[X]^2
$\text{Var}[ X] = {\mathbb E}[X^2]- {\mathbb E}[X]^2$

Theorem (Jensen inequality): Let $X$ be a random variable on a finite probability space and $\phi$ a convex function. Then

{\mathbb E}[\phi(X)]\geq \phi({\mathbb E}[X])
${\mathbb E}[\phi(X)]\geq \phi({\mathbb E}[X])$

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