1. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Chapter 5. Continuous Probability Distributions Sections 5.4, 5.5: Exponential and Gamma Distributions Jiaping Wang Department of Mathematical Science 03/25/2013, Monday

2. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Outline Exponential: PDF and CDF Exponential: Mean and Variance Gamma: PDF and CDF Gamma: Mean and Variance More Examples

3. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 1. Part 1. Exponential: PDF and CDF

4. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Probability Density Function θ = 2 θ = 1/2

5. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Cumulative Distribution Function θ = 2 θ = 1/2

6. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 2. Mean and Variance

7. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Gamma Function

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

9. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 5.9 A sugar refinery has three processing plants, all of which receive raw sugar in bulk. The amount of sugar that one plant can process in one day can be modeled as having an exponential distribution with a mean of 4 tons for each of the three plants. If the plants operate independently, find the probability that exactly two of the three plants will process more than 4 tons on a given day.

10. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Consider a particular plant in Example 5.9. How much raw sugar should be stocked for that plant each day so that the chance of running out of product is only 0.05? Example 5.10

11. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Properties

12. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 3. Part 3. Gamma: PDF

13. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Probability Density Function (PDF)

14. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Part 4. Mean and Variance

15. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

16. Example 5.11 A certain electronic system has a life length of X1, which has an exponential distribution with a mean of 450 hours. The system is supported by an identical backup system that has a life length of X2. The backup system takes over immediately when the system fails. If the system operate independently, find the probability distribution and expected value for the total life length of the primary and backup systems.

17. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Example 5.12 Suppose that the length of time X needed to conduct a periodic maintenance check on a pathology lab’s microscope (known from previous experience) follows a gamma distribution with α=3 and β=2 (minutes). Suppose that a new repairperson requires 20 minutes to check a particular microscope. Does this time required to perform a maintenance check seem our of line with prior experience? Answer: so μ=E(X)=αβ=6, σ 2 =V(X)=αβ 2 =12, the standard deviation σ=3.446, When x=20 minutes required from the repairperson, the deviation is 20-6=14 minutes, Which exceeds the mean 6 by k=14/3.446 standard deviations, so based on the Tschebysheff’s inequality, we have P(|X-6|≥14)≤(3.446/14) 2 =0.06, which is really small Probability, so we can say it is out of line with prior experience.

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

19. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 1 An insurance policy reimburses dental expense, X, up to a maximum benefit of 250. The probability density function for X is: where c is a constant. Calculate the median benefit for this policy. An insurance policy reimburses dental expense, X, up to a maximum benefit of 250. The probability density function for X is: where c is a constant. Calculate the median benefit for this policy. Answer: If P(X>a)=1/2, then a is a median. So c=250. As F(x)=1-exp(-x/250), we have 1-exp(-x/250)=1/2  x=250[ln(2)] = 173.29

20. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 2 Let X be an exponential random variable such that P(X>2) = 2P(X>4). Find the variance of X. Let X be an exponential random variable such that P(X>2) = 2P(X>4). Find the variance of X.

21. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Additional Example 3 If X has probability density function given by Find the mean and variance. If X has probability density function given by Find the mean and variance. Answer: Change it to the standard form with α=3, β=/12, so we can find E(X)=αβ=3/2, V(X)=αβ 2 =3/4.