The mean of a random variable, X, is the weighted average of the possible values of the random variable; each of the possible values is weighted by its probability. Two examples will illustrate the idea.
Discrete random variables
Suppose that the possible values for a random variable X are {1,2,3}, and P(X=1) = 1/2, P(X=2) = 1/4 and P(X=3) = 1/4. The mean of the random variable is given by:
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More generally, if the sample space for a discrete random variable is a set of values {xj, j=1,...,J} with probabilities given by pj, then the mean of X is given by:
Consider the example of high school grade point averages (GPAs) for first year college students in 2007. Let N represent the number of first year students. Then one way to write the sample space is to list the values of the GPA for each student, {x1,x2,...,xN}, and the probability of each element is 1/N. The mean is then the sum of all of the values divided by N.
Another way to write the sample space is to list the possible values for the GPA, [ 0.00,0.01,...,1.00,1.01,...,3.99,4.00]. The probability for each value is the proportion of students who obtained that particular GPA. Specifically, P(4.00) is the proportion of all students who achieved a 4.00 GPA. The mean is calculated using the previous formula.
Continuous random variables
Suppose the sample space for a random variable X is the interval [a,b], and the probability density function for X is f(X ). Then the mean of X is given by: