EXPECTED VALUE
Introduction
For an experiment or general random process, the outcomes are never fixed. We may replicate the
experiment and generally expect to observe many different outcomes. Of course, in most reasonable
circumstances we will expect these observed differences in the outcomes to collect with some level of
concentration about some central value. One central value of fundamental importance is the expected
value.
The expected value or expectation (also called the mean) of a random variable X is the weighted
average of the possible values of X, weighted by their corresponding probabilities. The expectation of
a random variable X is the value of X that we would expect to see on average after repeated
observation of the random process.
Definition:
Expected Value of a Discrete Random Variable
The expected value, , of a random variable X is weighted average of the possible values of X,
weight by their corresponding probabilities:
Where, N is the number of possible values of X.
Note:
Do not confuse the expected value with the average value of a set of observations: they
are two different but related quantities. The average value of a random variable X would be just
the ordinary average of the possible values of X. The expected value of X is a weighted average,
where certain values get more or less weight depending on how likely or not they are to be
observed. A true average value is calculated only when all weights (so all probabilities) are the
same.
The definition of expected value requires numerical values for the xk. So if the outcome for an
experiment is something qualitative, such as "heads" or "tails", we could calculate the expected
value if we assign heads and tails numerical values (0 and 1, for example).
Example:
Grade Distributions
Suppose that in a class of 10 people the grades on a test are given by 30, 30, 30, 60, 60, 80, 80, 80, 90,
100. Suppose a test is drawn from the pile at random and the score X is observed.
1. Calculate the probability density function for the randomly drawn test score.
2. Calculate the expected value of the randomly drawn test score.
Solution
Looking at the test scores, we see that out of 10 grades,
the grade 30 occurred 3 times
the grade 60 occurred 2 times
the grade 80 occurred 3 times
the grade 90 occurred 1 time
the grade 100 occurred 1 time
This tells us the Probability Density Function (PDF) of the randomly chosen test score X which we
present formally in the following table.
Grade, xk Probability, Pr(X = xk )
30 3/10
60 2/10
80 3/10
90 1/10
100 1/10
The expected value of the random variable is given by the weighted average of its values:
The PDF Could Be Presented as a Graph
The PDF was listed in a table, but an equivalent representation could be given in a graph that plots the
possible outcomes on the horizontal axis, and the probabilities associated to these outcomes on the
vertical axis.
We have added points where the probability is zero (test scores of 0, 10, 20, 40, 50, 70). It isn't
necessary to have these points displayed, but having these points on a graph of a PDF can often add
clarity.
Notice that the expected value of our randomly selected test score, = 64, lies near the "centre"
of the PDF. There are many different ways to quantify the "centre of a distribution" - for example,
computing the 50th percentile of the possible outcomes - but for our purposes we will concentrate our
attention on the expected value.
Reference:
http://wiki.ubc.ca/Science:MATH105_Probability/Lesson_1_DRV/1.04_The_Expected_Value_of_a_Discrete_PDF
