Random Variables

 

	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 E(a)=a, 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 , . 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).