Random Variables, Distributions, and Expected Value

1. A random variable is a variable that takes specific values with specific probabilities. It can be thought of as a variable whose value depends on the outcome of an uncertain event. 2. We usually denote random variables by capital letters near the end of the alphabet; e.g., X,Y,Z. 3. Example: Let X be the outcome of the roll of a die. Then X is a random variable. Its possible values are 1, 2, 3, 4, 5, and 6; each of these possible values has probability 1/6. 4. The word “random” in the term “random variable” does not necessarily imply that the outcome is completely random in the sense that all values are equally likely. Some values may be more likely than others; “random” simply means that the value is uncertain. 5. When you think of a random variable, immediately ask yourself • What are the possible values? • What are their probabilities? 6. Example: Let Y be the sum of two dice rolls. • Possible values: {2, 3, 4, . . . , 12}. • Their probabilities: 2 has probability 1/36, 3 has probability 2/36, 4 has probability 3/36, etc. (The important point here is not the probabilities themselves, but rather the fact that such a probability can be assigned to each possible value.) 7. The probabilities assigned to the possible values of a random variable are its distribution. A distribution completely describes a random variable.