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Lecture 9. Continuous Probability Distributions

Lecture 9. Continuous Probability Distributions. David R. Merrell 90-786 Intermediate Empirical Methods for Public Policy and Management. Agenda. Normal Distribution Poisson Process Poisson Distribution Exponential Distribution. Continuous Probability Distributions.

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Lecture 9. Continuous Probability Distributions

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  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) 4/8 3/8 2/8 1/8 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: Yc2>Xc1 • Click: Display > Connect > OK

  16. Normal Curve Output

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

  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: Yc2>Xc1 • 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

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