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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

Lecture 9. Continuous Probability Distributions

David R. Merrell

90-786 Intermediate Empirical Methods for Public Policy and Management

agenda
Agenda
  • Normal Distribution
  • Poisson Process
  • Poisson Distribution
  • Exponential Distribution
continuous probability distributions
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
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
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
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
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
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
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
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
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
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
Poisson Process

rate

x

x

x

time

0

Assumptions

time homogeneity

independence

no clumping

poisson process1
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
Count: Poisson Distribution
  • What is the probability that 3 earthquakes will strike during the next six months?
poisson distribution
Poisson Distribution

Count in time period t

minitab probability calculation
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
Duration: Exponential Distribution
  • Time between occurrences in a Poisson process
  • Continuous probability distribution
  • Mean =1/t
exponential probability problem
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 calculation1
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
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
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