1. Uniform Distributions and Random Variables Lecture 23 Sections 6.3.2, 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. 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. A uniform distribution must have two endpoints. Call them a and b. Call them a and b. The graph of the uniform random variable: The graph of the uniform random variable: a b

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

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

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

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

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

9. Waiting Times A traffic light at an intersection stays red for 30 seconds. A traffic light at an intersection stays red for 30 seconds. Cars appear at the intersection at random times. 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. 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. Let X be the waiting time. What is the distribution of X? What is the distribution of X?

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

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

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

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

14. Waiting Times What proportion of the cars will wait at least 10 seconds? What proportion of the cars will wait at least 10 seconds? The proportion is 20/30, or 0.6667. The proportion is 20/30, or 0.6667. 0 30 1/30 10 Area = 0.6667 x

15. Waiting Times Can you think of a reason why the uniform model may not be appropriate for the situation described? 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. If X is a uniform variable on the interval [a, b], then the mean of X is the midpoint (a + b)/2. The variance of X is (b – a) 2 /12. The variance of X is (b – a) 2 /12. The standard deviation of X is (b – a)/  12. The standard deviation of X is (b – a)/  12. In the previous example, what is the average waiting time for the cars stopped by the red light? 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! 6.10, p. 386 – Three Distributions. Let’s Do It! 6.10, p. 386 – Three Distributions.