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Uniform Distributions and Random Variables

Explore the concept of uniform distributions, random variables, and probability functions in statistics. Learn about waiting times, mean calculations, and examples of discrete probability functions.

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Uniform Distributions and Random Variables

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

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

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

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

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

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

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

  8. 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

  9. 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?

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

  11. Waiting Times • What proportion of the cars will wait at least 10 seconds? 1/30 0 30

  12. Waiting Times • What proportion of the cars will wait at least 10 seconds? 1/30 0 10 30

  13. Waiting Times • What proportion of the cars will wait at least 10 seconds? 1/30 0 10 30

  14. 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 0 10 30

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

  16. 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?

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

  18. 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.

  19. 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.

  20. 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}.

  21. 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?

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

  23. 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.

  24. 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?

  25. Example of a Discrete PDF • We may present the pdf as a table.

  26. 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 0 1 2 3

  27. 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 0 1 2 3

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

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