1. 1 Random Variables and Probability Distributions: Discrete versus Continuous For this portion of the session, the learning objective is:  Learn that the world is not always “Normal”; that is, there are other probability distributions with which a manager should be familiar: Custom, Binomial, Uniform, and Triangular.

2. 2 Probability Distributions A random variable X is a value about which we are uncertain. There are two types of random variables: Discrete and Continuous and Discrete Random Variable X  Takes on one of a finite number of values. (NOTE: Technically, the range of values can be countably infinite, although this will not happen in this course.)  The uncertainty about the value of X is represented by a probability distribution p(x), where

3. 3 Example 1 of a Discrete Random Variable: A “Custom” Random Variable Value x Probability p(x) 80.10 120.25 180.30 200.20 250.15 Mean = (.10)(8) + (0.25)(12) + (0.30)(18) + (0.20)(20) + (0.15)(25) = 16.95 Note that, in this example, the mean is not a value that the random variable X can take on.

4. 4 Example 2 of a Discrete Random Variable: A Binomial Random Variable Not discussed in this session.

5. 5 Continuous Random Variable X  Takes on one of an infinite number of values.  The uncertainty about X is represented by a probability distribution f(x), where and

6. 6 Mean = 2 Example 1 of a Continuous Random Variable: A Normal Random Variable

7. 7 Mean = Min + Max = 8 2 Example 2 of a Continuous Random Variable: A Uniform Random Variable

8. 8 Mean = Min + Likeliest + Max = 12 3 Example 3 of a Continuous Random Variable: A Triangular Random Variable A Triangular Random Variable is often used in one the following situations:  When a Normal Random Variable cannot be used because the distribution is clearly skewed to the left or right.  When, in the absence of more complete data, it is easy to specify a worst case, a best case, and a likeliest (i.e., most-likely) case.