Expected value

Once we have constructed the probability distribution for a random variable, we often want to compute the mean or expected value of the random variable. The mean of discrete random variable X, denoted either

or , is actually the mean of its probability distribution. The mean

(or expected) value of a discrete random variable is the value that we expect to observe per repetition, on average, if we perform an experiment a large number of times. For example, we may expect a house salesperson to sell on average, 3.50 houses per month. It does not mean that every month this salesperson will sell exactly 3.50 houses. (Actually he (or she) can not sell exactly 3.50 houses). This simply means that if we observe for many months, this salesperson will sell a different number of houses different months. However, the average of all sold houses in these months will be 3.50.

Definition:

The mean (or expected value) of discrete random variable X is defined as

Here the sum extends over all distinct values x of X.

In order to compute the expected value of a discrete random variable we must multiply each value of the random variable by the corresponding value of its probability function. We then add the resulting terms.

Example:

Sales show that five is the maximum number of cars sold on a given day at car selling company. Table 3.4 shows probability distribution of cars sold per day. Find the expected number of cars sold.

Table 3.4

x	P (x)
	0.18 0.39 0.24 0.14 0.04 0.01

Solution:

To find the expected number (or mean) of cars sold, we multiply each value of x by its probability and add these results.

x	P (x)
	0.18 0.39 0.24 0.14 0.04 0.01	0.00 0.39 0.48 0.42 0.16 0.05

In fact, it is impossible for company to sell exactly 1.50 cars in any given day. But we examine selling cars at this company for many days into the future, and see that, the expected value of 1.50 cars provides a good estimate of the mean or average daily sales volume. The expected value can be important to the managers from both planning and decision making points of view.

For example, suppose that this company will be open 40 days during next

2-month. How many cars should the owner expect to be sold during this time?

While we can not specify the exact value of 1.50 cars, it provides an expected sale of cars for the next 2-month period.

Дата добавления: 2015-08-05