Expected Value of a Probability Distribution The
expected value of a random variable tells you the value that you would
expect to get if you measured the variable many times. The expected
value is also known as the mean of the distribution. The expected
value of a discrete random variable x is a weighted average of all the possible values of x. It is found by multiplying each of the possible outcomes of the random variable by the probability of that outcome.
m_{x} = the Greek letter m is the mean of the distribution
E(x) = the expected value of the random variable x
i = a possible outcome of the random variable x
n = the number of possible outcomes
x_{i} = the ith outcome of the random variable x
p_{i} = the probability of outcome x_{i}
The expected value of a discrete random variable does not have to be
one of the possible values of a random variable. For example, in three
flips of a fair coin, the expected value of the outcome heads is 1.5,
although "1.5" is not a possible value for the number of occurrences of
the random variable heads in three flips of a coin.
For a simple demonstration of calculating expected value, consider a
stock with two possible returns (depending on market conditions); the
probability of each return is described by the table below.
The expected return for this stock can be calculated using the formula
The expected return for this stock is 8.8 percent.
For a more challenging example of calculating expected value, return
to the earlier example of the number of orders placed per week from the
Discrete Random Variables portion of the Discrete Probability
Distributions section. Imagine that you are responsible for having
enough product on hand to fill all incoming orders. You want to know
the expected order volume so that you may plan accordingly. You may
recall the possible order values and the probability associated with
each value, as shown in the graph below.
To calculate the expected value of x, use the following formula.
Because there are a large number of values in this example, the
calculation is shown in a table below. The steps for the calculation
are as follows.
Multiply each value of x by the probability of its occurrence (p). In the table below, the result of this calculation is displayed in the column labeled x_{i} * p_{i}.
Sum all of the values of x weighted by the probability of their occurrences. This is shown in the final row of the table labeled Sum.
The expected value of the number of orders per week is 44.61.
Measures of Dispersion of a Probability Distribution
While the expected value does provide more information about the random
variable of interest, it does not tell the whole story. Measures of
dispersion—variance and standard deviation—indicate how close the
actual value may fall to the expected value. For example, consider the
following two investments.
If you calculated the expected value of the investments, you would
find that it is zero in both cases. This might lead you to believe that
the returns on the investments are very similar, but, as you can see
from the graphs, they are actually quite different. The measures of
dispersion—variance and standard deviation—would tell you that the
actual return for the first investment will be far from the expected
value, while the actual return for the second investment will be closer
to the expected value.
Variance of a Probability Distribution Variance is a
measure that describes the distance that the actual value might fall
from the expected value. The variance of a probability distribution is
calculated by determining the distance between each of the possible
values for a random variable and the expected value. This distance may
be positive or negative, as the possible values of the random variable
may fall above or below the expected value. The distance is squared to
avoid the issue of positive and negative numbers canceling each other
out. The squared distances are then multiplied by the probability of
the outcome and summed. The variance is simply the weighted average of
the squared distances of each outcome from the mean where the weights
are the probabilities of each outcome. The formula to calculate the
mean of a probability distribution is
VAR(x) and s_{x}^{2} represent the variance of the distribution of random variable x
i = a possible outcome of the random variable x
n = the number of possible outcomes
x_{i} = the ith outcome of random variable x
p_{i} = the probability of outcome x_{i}
m_{x} = the expected value of random variable x
Return to the stock return example for a demonstration of variance.
You may recall the possible outcomes and probabilities, as displayed in
the following table.
Earlier, you calculated the expected return to be 8.8 percent. Now
you can use the formula for variance of a discrete probability
distribution to calculate the variance.
The variance of the stock's return is 2.56.
You can also apply the variance formula to the weekly order volume
example. The steps are listed below and the calculations are shown in
the following table.
Find the difference from each possible value to the expected value
of the random variable. This is shown in column 3 of the table below.
(Remember, m_{x} = 44.61.)
Square the difference and multiply it by the probability of that
outcome. You can see the result of this calculation in column 4.
Sum the squared differences times the probabilities to get the variance. This figure is displayed at the bottom of column 4.
The variance of weekly order volumes is 2.99 orders squared. A
larger value of the variance indicates that there is a greater chance
the actual value of a random variable will fall far from the mean.
