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).