The Expected Values and Simulations module targets the following cognitive tasks:
| Task Skills | Concepts | |
|---|---|---|
| Exp-1: | Understand the expected value of a random variable | |
| Exp-2: | Calculate the expected value of a discrete numeric random variable | Understand the expected value of a discrete numeric random variable |
| Exp-3: | Simulate the experimental expected value of a distrete numeric random variable | Understand the experimental expected value of a distrete numeric random variable |
| Exp-4: | Understand the relation between the theoretical and experimental expected values | |
| Exp-5: | Understand the properties of expected values | |
| Exp-6: | Calculate the variance of a discrete numeric random variable | Understand the variance of a discrete numeric random variable |
| Exp-7: | Simulate the experimental variance of a distrete numeric random variable | Understand the experimental variance of a distrete numeric random variable |
The expected value of a random variable, denoted by , is the balance point of its probability distribution, i.e, the point at which the distribution would balance on a fulcrum. The expected value of a random variable is also called the theoretical expected value. The expected value is often used to compute the expected gain or loss from playing a game of chance.
The expected value of is equivalent to the theoretical mean of
, i.e,
where
is the theoretical mean. Conceptually, the theoretical mean is the average of a long sequence of realizations of the random variable
. This idea will be discussed more fully in Exp-3 below.
Consider the discrete random variable which takes on values
with corresponding probabilities
. The theoretical mean is defined by:
The five-step method is implemented in the following Java applet:
Example #1 computes the experimental expected number of cereal boxes that must be bought to obtain all six colored pens.
Example #2 computes the experimental expected number of girls in a three-child family.
Self-test