A3.1.1 Sample space and probability measure

A sample space ? is the set of all possible outcomes (or events) of an experiment. An outcome A is a subset of ?. A probability measure Pr( ) is a mapping from ? to the real line with the following properties.

1. Pr( A) ? 0, A ? ?;

2. Pr( ?) = 1;

3. For a countable set of events, { A _m}, if A _l ? A _m = , for l ? m, then

   
   

A3.1.2 Joint and conditional probability

The joint probability of two events A and B is Pr( A ? B). The conditional probability of A given B is given by

Two events A and B are independent if and only if

and this implies that

A3.1.3 Random variables

A random variable is a mapping from an event ? in ? to a real number, denoted by X( ?). The cumulative distribution function (cdf) of X is defined by

and the probability density function (pdf) is defined by

where the subscript X on F and f identifies the random variable. If the random variable is obvious, the subscript is often omitted.

There are some well known pdfs as follows:

1. Gaussian pdf with mean and variance ? ²:

   
   

2. expondential pdf ( a >...