Definitions

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Probability

The sample space (denoted $S$) is defined as the set of all possible distinct outcomes or “events” of an experiment. Using this perspective, a classic idea about ‘probability’ is

$$\frac{ \text{number of ways the event can occur} }{ \text{number of outcomes in S }}$$

provided all points in $S$ are equally likely.

Definition Let $S = \{a_1, a_2, a_3, ...\}$ be a discrete sample space. Then the probabilities $P(a_i)$ are numbers attached to the $a_i\text{'s}$ such that two conditions hold:

(1) $0 \leq P(a_i) \leq1$

(2) $\sum\limits_{i} P(a_i) = 1$

Definition An event in a discrete sample space is a subset $A\subset S$. If the event contains only one point, e.g. $A_1 = \{a_i\}$ we call it a simple event. An event A made up of two or more simple events, e.g. $A_1 = \{a_1, a_2\}$ is a compound event.

Definition The probability $P(A)$ of an event $A$ is the sum of the probabilities for all the simple events that make up $A$.

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Random Variables

A random variable is a numerical-valued variable that represents the outcomes in an experiment or random process. Typically, random variables are denoted by an upper-case letter such as $X$. The corresponding lower case letter is often reserved to refer to one of a number of possible values that the random variable can take on. For example, if a coin is tossed 3 times then

$$X=\text{Number of Heads}$$ $$x=\{1, 2, 3\}$$

Definition A random variable is a function that assignes a real number to each point in a sample space $S$.

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Probability Function

One type of desired description of a random variable is a summary for how probability is distributed amongst the possible values a random variable can take on.

Definition The probability function of a random variable $X$ is the function

$$f(x)=P(X=x)\text{, for all x in the domain A }$$

The pairs $\{(x, f(x)):x \in A\}$ is collectively the probability distribution. All probability functions share two properties:

(1) $f(x)\geq0$

(2) $\sum_\limits{x \in A}f(x)=1$

Distribution Function

An alternative to the probability distribution for describing a probability model is the cumulative distribution function or simply the distribution function.

Definition The distribution function of a random variable $X$ is

$$F(x)=P(X\leq x)\text{, for all x in the domain A }$$

All distribution functions share three properties:

(1) $F(x)\text{ is a non decreasing function of x}$

(2) $0\leq F(x) \leq 1 \text{ for all x }$

(3) $\lim_{x\to-\infty} F(x)=0 \text{ and } \lim_{x\to+\infty} F(x)=1$