Random Variables

In an experiment, each outcome can be associated with a single number by using a rule of association, or random variable. By converting outcomes to numbers, we can use mathematics to give us insight into the experiment or study being conducted.

Variable A rule which associates a single number with the outcome of an experiment (an event).
Random Variable The random variable takes a countable number of values. In other words, it has a discrete sample space.
Random Variable The random variable takes on values in an interval (it has a continuous sample space).

NOTE:\ We usually denote random variables with capital letters, e.g., the random variables X or Y.

EXAMPLE:\ Consider the following random variables:

X = 1 (0) if a randomly selected person prefers Pepsi (Coke). (Categorical random variable.)
X = # of Aggies attending a home football game. (Discrete random variable.)
X = length of a rainbow trout. (Continuous random variable.)

Mean and Variance of Random Variables
Mean and Variance of Random Variables
- The mean [or expected value] of a discrete random variable X, denoted [or E(X)], is given by
  
  NOTE: (Properties of E): Let a and b be real constants, then
  1. E(a)=a,
  2. E(aX+b)=aE(X)+b.
- The variance of a discrete random variable X, denoted [or ], is defined as
  
  A shortcut formula for the variance, which is usually easier to evaluate then the definition formula, is
  
  or in symbols,
  
  NOTE: (Properties of ): Let a and b be real constants, then
  1. ,
  2. .
EXAMPLE:\ At a certain country market, the probabilities (long-run frequencies) for the number of apples bought by customers was tabulated:

The expected value of the number of apples purchased is

Its variance is computed in two stages: first we calculate as follows,

and then we apply the formula for the variance

The standard deviation of X is obtained by .
NOTE:\ The expected value of a random variable X is just a weighted average of the values of X, where the weights are the associated probabilities (or relative frequencies).