Uniform Distributions and Random Variables Lecture 23 Section 7.5.1 Mon, Oct 25, 2004

A Note About Probability The probability that something happens is the proportion of the time that it does happen out of all the times it was given an opportunity to happen. Therefore, “probability” and “proportion” are synonymous in the context of what we are doing.

Examples of Random Variables Roll two dice. Let X be the number of sixes. Possible values of X = {0, 1, 2}. Roll two dice. Let X be the total of the two numbers. Possible values of X = {2, 3, 4, …, 12}. Select a person at random and give him up to one hour to perform a simple task. Let X be the time it takes him to perform the task. Possible values of X are {x | 0 ≤ x ≤ 1}.

Types of Random Variables Discrete Random Variable – A random variable whose set of possible values is a discrete set. Continuous Random Variable – A random variable whose set of possible values is a continuous set. In the previous examples, are they discrete or continuous?

Discrete Probability Distribution Functions Discrete Probability Distribution Function (pdf) – A function that assigns a probability to each possible value of a discrete random variable.

Example of a Discrete PDF Roll two dice and let X be the number of sixes. Draw the 6  6 rectangle showing all 36 possibilities. From it we see that P(X = 0) = 25/36. P(X = 1) = 10/36. P(X = 2) = 1/36. (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

Example of a Discrete PDF Suppose that 10% of all households have no children, 30% have one child, 40% have two children, and 20% have three children. Select a household at random and let X = number of children. Then X is a random variable. Which step in the procedure is left to chance? What is the pdf of X?

Example of a Discrete PDF We may present the pdf as a table. x P(X = x) 0.10 1 0.30 2 0.40 3 0.20

Example of a Discrete PDF Or we may present it as a stick graph. P(X = x) 0.40 0.30 0.20 0.10 x 1 2 3

Example of a Discrete PDF Or we may present it as a histogram. P(X = x) 0.40 0.30 0.20 0.10 x 1 2 3

Let’s Do It! Let’s do it! 7.20, p. 426 – Sum of Pips.