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

Uniform Distributions Uniform distribution – A continuous distribution in which all values within a given range are equally represented in the population.

Uniform Distributions A uniform distribution must have two endpoints. Call them a and b. The graph of the uniform variable: a b

Uniform Distributions A uniform distribution must have two endpoints. Call them a and b. The graph of the uniform variable: a b

Uniform Distributions A uniform distribution must have two endpoints. Call them a and b. The graph of the uniform variable: Area? a b

Uniform Distributions A uniform distribution must have two endpoints. Call them a and b. The graph of the uniform variable: Area = 1 a b

Uniform Distributions A uniform distribution must have two endpoints. Call them a and b. The graph of the uniform variable: ? Area = 1 a b

Uniform Distributions A uniform distribution must have two endpoints. Call them a and b. The graph of the uniform variable: 1/(b – a) Area = 1 a b

Waiting Times A traffic light at an intersection stays red for 30 seconds. Cars appear at the intersection at random times. For each car that gets stopped by a red light, we observe how long it waits until the light turns green. Let X be the waiting time. What is the distribution of X?

Waiting Times In the simplest model, X has a uniform distribution from 0 sec to 30 sec. 1/30 30

Waiting Times What proportion of the cars will wait at least 10 seconds? 1/30 30

Waiting Times What proportion of the cars will wait at least 10 seconds? 1/30 10 30

Waiting Times What proportion of the cars will wait at least 10 seconds? 1/30 10 30

Waiting Times What proportion of the cars will wait at least 10 seconds? The proportion is 20/30, or 0.6667. 1/30 Area = 0.6667 10 30

Waiting Times Can you think of a reason why the uniform model may not be appropriate for the situation described?

The Mean of a Uniform Variable If X is a uniform variable on the interval [a, b], then the mean of X is the midpoint (a + b)/2. In the previous example, what is the average waiting time for the cars stopped by the red light?

Let’s Do It! Let’s Do It! 6.13, p. 357 – Three Distributions.

Random Variables Random variable – A variable whose value is determined by the outcome of a procedure. The procedure includes at least one step whose outcome is left to chance. Therefore, the random variable takes on a new value each time the procedure is performed.

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.