Section C.2. Random Variables

C.2. Random Variables

Sometimes, the behavior of outcomes on a sample space whose values are real numbers must be considered . A random variable , X(‰) , or simply X , is a function that maps each outcome of the sample space to real numbers. In the previous example, 0 and 1 are real numbers, but they could have been interpreted as "success" and "fail" as outcomes. In such a case, the random variable maps them into 0 and 1, respectively.

C.2.1. Basic Functions

The probability that a random variable, X , takes a value that does not exceed a given number, x , is called the cumulative distribution function (CDF) of X . This is a function of x and is usually denoted by F _X (x) :

Equation C.6

Random variables are either discrete or continuous . For a discrete random variable, we define, the probability mass function (PMF) to be the probability function of the random variable at any given number x . PMF is denoted by

Equation C.7

It is obvious that for a discrete random variable X , CDF at any given point x is computed by adding all PMF values up to point x :

Equation C.8

Similarly, we can define the probability density function (PDF) for a continuous random variable to be the probability function of X . PDF is denoted by f _X (x) . Similarly, CDF and PDF are associated with each other through

Equation C.9

C.2.2. Conditional Functions

We can define the conditional CDF of a random variable X as a CDF given that an event A has already occurred. This CDF is obtained naturally by

Equation C.10

Similarly, the conditional PDF can be defined as

Equation C.11

and the conditional PMF can be defined as

Equation C.12

C.2.3. Popular Random Variables

Three popular discrete random variables are Bernulli , binomial , and Poisson random variables. Three popular continuous random variables are uniform , Gaussian , and exponential . All are briefly reviewed.

Bernulli Random Variable

Bernulli random variable X is discrete and is defined over a sample space S _X ={0, 1}, where 0 and 1 represent failure with probability p and success with probability 1 - p , respectively. Therefore, PMF of this random variable is defined by

Equation C.13

Binomial Random Variable

A binomial random variable X is discrete and is defined over a sample space S _X = {0, 1, ... , n }. This random variable is basically n Bernulli random variables, and its PMF is obtained from

Equation C.14

Geometric Random Variable

A geometric random variable X is discrete and is defined over a sample space S _X = {1, 2, ... , x }. This random variable is defined for counting x Bernulli random trials in a binomial random variable with only one success at the last trial. Consequently, its PMF is obtained from

Equation C.15

Poisson Random Variable

A Poisson random variable X is discrete and is defined over a sample space S _X = {1,2, ...}. This random variable is approximated for a binomial random variable when n large and p is small. With these conditions, its PMF is obtained from Equation (C.14), taking into account the mentioned approximations:

Equation C.16

Uniform Random Variable

A uniform random variable X is continuous and is defined over a sample space S _X = [ a , b ], where a and b are two constant numbers. PDF of a uniform random variable is obtained from

Equation C.17

Exponential Random Variable

An exponential random variable X is continuous and is defined over a sample space S _X = [ o , ). PDF of an exponential random variable is expressed by

Equation C.18

Gaussian (Normal) Random Variable

A Gaussian (normal) random variable X is continuous and is defined over a sample space S _X = [- , + ). PDF of an exponential random variable is expressed by

Equation C.19

where E [ X ] and V [ X ] are the expected value and the variance of the random variable, respectively.

C.2.4. Expected Value and Variance

For a random variable X , the expected value , or mean, E [ X ], is defined as the statistical average of all possible values of the random variable. Thus, for a discrete random variable X taking on values from a total of N possible values, the expected value is

Equation C.20

The concept is identical for a continuous random variable having infinite points:

Equation C.21

We can also define the variance of a random variable that gives a measure of how values differ :

Equation C.22

C.2.5. A Function of Random Variable

If g(X) is a function of X , the expected value of g(X) for a discrete random variable can also be defined as follows :

Equation C.23

and for a continuous random variable:

Equation C.24

The expected value of a random variable by itself may not be useful for the numerical evaluation of the random variable. The reason is that two random variables with an identical expected value can take on values, each from a totally different range of numbers. The variance can be calculated by using either Equation C.23 or C.24, depending on the type of the random variable.