Lecture 9. Continuous Probability Distributions

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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

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
• 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

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
Normal Distributions
• Why so important?
• Many statistical methods are based on the assumption of normality
• Many populations are approximately normally distributed
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!
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 - ) / 

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
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?
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?
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

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
Poisson Process

rate

x

x

x

time

0

Assumptions

time homogeneity

independence

no clumping

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

Count in time period t

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

Duration: Exponential Distribution
• Time between occurrences in a Poisson process
• Continuous probability distribution
• Mean =1/t
Exponential Probability Problem
• What is the probability that 9 months will pass with no earthquake?
• t = 1/12 = 1/3
• 1/ t = 3
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

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
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