1. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Chapter 5. Continuous Probability Distributions Sections 5.2, 5.3: Expected Value of Continuous Random Variables and Uniform Distribution Jiaping Wang Department of Mathematical Science 03/20/2013, Monday

2. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Outline EV: Definitions and Theorem EV: Examples Uniform Distribution: Density and Distribution Functions Uniform Distribution: Mean and Variance More Examples Homework #8

3. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 1. Part 1. EV: Definitions and Theorem

4. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Definition and Theorem

5. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Variance

6. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 2. EV: Examples

7. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 5.4

8. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

9. Example 5.5

10. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

11. Tchebysheff’s Theorem and Example 5.6 The Tchebysheff’s theorem holds for the continuous random variable, X, ie., P(|X-μ|<kσ) ≥ 1-1/k 2 Example 5.6: The weekly amount X spent for chemicals by a certain firm has a mean of $1565 and a variance of $428. Within what interval should these weekly costs for chemicals be expected to lie in at least 75% of the time? Answer: To find the interval guaranteed to contain at least 75% of the probability mass for X, we need to have 1-1/k 2 =0.75  k=2. So the interval is given by [1565-2(428) 1/2, 1565+2(428) 1/2 ]. The Tchebysheff’s theorem holds for the continuous random variable, X, ie., P(|X-μ|<kσ) ≥ 1-1/k 2 Example 5.6: The weekly amount X spent for chemicals by a certain firm has a mean of $1565 and a variance of $428. Within what interval should these weekly costs for chemicals be expected to lie in at least 75% of the time? Answer: To find the interval guaranteed to contain at least 75% of the probability mass for X, we need to have 1-1/k 2 =0.75  k=2. So the interval is given by [1565-2(428) 1/2, 1565+2(428) 1/2 ].

12. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 3. Part 3. Uniform Distribution: Density and Distribution Functions

13. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Density Function

14. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Cumulative Distribution Function

15. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Mean and Variance

16. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 5.7 A farmer living in western Nebraska has an irrigation system to provide water for crops, primarily corn, on a large farm. Although he has thought about buying a backup pump, he has not done so. If the pump fails, delivery time X for a new pump to arrive is uniformly distributed over the interval from 1 to 4 days. The pump fails. It is a critical time in the growing season in that the yield will be greatly reduced if the crop is not watered within the next 3 days. Assuming that the pump is ordered immediately and the installation time is negligible, what is the probability that the farmer will suffer major yield loss?

17. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 3. More Examples

18. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 1 Let X have the density function given by 1.Find the value c. 2.Find F(x). 3.P(0≤X≤0.5). 4.E(X). Let X have the density function given by 1.Find the value c. 2.Find F(x). 3.P(0≤X≤0.5). 4.E(X).

19. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 2 Let X have the density function Find E(lnX). Let X have the density function Find E(lnX).

20. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Homework #8 Page 199-200: 5.4, 5.7 Page 209: 5.22 Page 214-215: 5.28, 5.40. Page 199-200: 5.4, 5.7 Page 209: 5.22 Page 214-215: 5.28, 5.40. Additional Hw1: Let X have the density function Find the E(X). Additional Hw2: The density function of X is given by (a). Find a and b. (b). Determine the cumulative distribution function F(x).