1. Lecture 9. Continuous Probability Distributions David R. Merrell 90-786 Intermediate Empirical Methods for Public Policy and Management

2. Agenda Normal Distribution Poisson Process Poisson Distribution Exponential Distribution

3. Continuous Probability Distributions Random variable X can take on any value in a continuous interval Probability density function: probabilities as areas under curve Example: f(x) = x/8 where 0  x  4 Total area under the curve is 1 P(x) 1/8 2/8 3/8 4/8 x

4. Calculations Probabilities are areas P(x < 1) is the area to the left of 1 (1/16) P(x > 2) is the area to the right of 2, i.e., between 2 and 4 (1/2) P(1 < x < 3) is the area between 1 and 3 (3/4) In general P(x > a) is the area to the right of a P(x < 2) = P(x  2) P(x = a) = 0

5. Normal Distributions Why so important? Many statistical methods are based on the assumption of normality Many populations are approximately normally distributed

6. Characteristics of the Normal Distribution The graph of the distribution is bell shaped; always symmetric The mean = median =  The spread of the curve depends on , the standard deviation Show this!

7. The Shape of the Normal and σ

8. Standard Normal Distribution Normal distribution with  = 0 and  = 1 The standard normal random variable is called Z Can standardize any normal random variable: z score Z = (X -  ) / 

9. Calculating Probabilities Table of standard normal distribution PDF template in Excel Example: X normally distributed with  = 20 and  = 5 Find: Probability that x is more than 30 Probability that x is at least 15 Probability that x is between 15 and 25 Probability that x is between 10 and 30

10. Percentages of the Area Under a Normal Curve Show this!

11. Percentages of the Area Under a Normal Curve

12. Example 1. Normal Probability An agency is hiring college graduates for analyst positions. Candidate must score in the top 10% of all taking an exam. The mean exam score is 85 and the standard deviation is 6. What is the minimum score needed? Joe scored 90 point on the exam. What percent of the applicants scored above him? The agency changed its criterion to consider all candidates with score of 91 and above. What percent score above 91?

13. Example 2. Normal Probability Problem The salaries of professional employees in a certain agency are normally distributed with a mean of $57k and a standard deviation of $14k. What percentage of employees would have a salary under $40k?

14. Minitab for Probability Click: Calc > Probability Distributions > Normal Enter: For mean 57, standard deviation 14, input constant 40 Output: Cumulative Distribution Function Normal with mean = 57.0000 and standard deviation = 14.0000 x P( X <= x) 40.0000 0.1123

15. Plotting a Normal Curve MTB > set c1 DATA > 15:99 DATA > end Click: Calc > Probability distributions > Normal > Probability density > Input column Enter: Input column c1 > Optional storage c2 Click: OK > Graph > Plot Enter: Y c2 > X c1 Click: Display > Connect > OK

16. Normal Curve Output

17. Poisson Process time homogeneity independence no clumping rate xxx 0 time Assumptions

18. Poisson Process Earthquakes strike randomly over time with a rate of = 4 per year. Model time of earthquake strike as a Poisson process Count: How many earthquakes will strike in the next six months? Duration: How long will it take before the next earthquake hits?

19. Count: Poisson Distribution What is the probability that 3 earthquakes will strike during the next six months?

20. Poisson Distribution Count in time period t

21. Minitab Probability Calculation Click: Calc > Probability Distributions > Poisson Enter: For mean 2, input constant 3 Output: Probability Density Function Poisson with mu = 2.00000 x P( X = x) 3.00 0.1804

22. Duration: Exponential Distribution Time between occurrences in a Poisson process Continuous probability distribution Mean =1/ t

23. Exponential Probability Problem What is the probability that 9 months will pass with no earthquake?  t = 1/12 = 1/3 1/ t = 3

24. Minitab Probability Calculation Click: Calc > Probability Distributions > Exponential Enter: For mean 3, input constant 9 Output: Cumulative Distribution Function Exponential with mean = 3.00000 x P( X <= x) 9.0000 0.9502

25. Exponential Probability Density Function MTB > set c1 DATA > 0:12000 DATA > end Let c1 = c1/1000 Click: Calc > Probability distributions > Exponential > Probability density > Input column Enter: Input column c1 > Optional storage c2 Click: OK > Graph > Plot Enter: Y c2 > X c1 Click: Display > Connect > OK

26. Exponential Probability Density Function

27. Next Time: Random Sampling and Sampling Distributions Normal approximation to binomial distribution Poisson process Random sampling Sampling statistics and sampling distributions Expected values and standard errors of sample sums and sample means