Understanding the entire set of outcomes associated with a random variable, and the probability of each outcome, may not be concise enough information for business decision-making purposes. Managers must rely on summary measures to support their decisions; summary measures identify the expected outcome as well as the expected level of variability around that outcome.

Summation

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.

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_x² 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.

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.

Standard Deviation of a Probability Distribution

Because the difference between the possible outcomes and the expected value of a random variable is squared in the calculation of variance, the resulting variance is presented in terms of units squared. For comparative purposes, the measure of dispersion needs to be in the same unit measure as the original data. Standard deviation fills this need. Standard deviation is a second measure of the level of dispersion of a random variable. It is simply the square root of the variance. The formula for standard deviation is

The standard deviation of weekly order volumes can be calculated as follows.

The standard deviation of weekly orders is 1.73 orders. Standard deviation also describes how far the actual value of a random variable is likely to fall from the expected value. Standard deviation is helpful in understanding the behavior of random variables, especially the degree to which they deviate from the mean as well as their level of volatility. These qualities will be explored in detail in later sections of this course.

1. A coffee shop relies on low prices for coffee beans to keep its margins and profits high. Coffee bean prices depend on the weather conditions during the coffee bean growing season. The following table provides the coffee-growing conditions and associated probabilities, along with the profits to the coffee shop under each scenario. Calculate the expected profit of the coffee shop.

Solution 1

2. A consumer products company recognizes that its revenue growth depends on successful new product launches. The following table shows the probability of success associated with the next new product launch and the associated revenues. Calculate the expected revenues from this new product, as well as the variance and standard deviation of those revenues.

Solution 2

3. A portfolio contains three investment instruments with the following dollar values of investment in each.

The return of each of these investment instruments over the next year is dependent on the conditions of the market. Five market conditions are possible with the following associated probabilities.

Using this table of the expected rate of return for each investment instrument at various market conditions, calculate the expected return, variance, and standard deviation on this portfolio.

Solution 3