Download 90-786 Lecture 9

Lecture 9. Continuous
Probability Distributions
David R. Merrell
90-786 Intermediate Empirical
Methods for Public Policy and
Management
Agenda
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Normal Distribution
Poisson Process
Poisson Distribution
Exponential Distribution
Continuous Probability Distributions
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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
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Probabilities are areas
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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
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P(x > a) is the area to the right of a
P(x < 2) = P(x  2)
P(x = a) = 0
Normal Distributions
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Why so important?
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Many statistical methods are based on the
assumption of normality
Many populations are approximately
normally distributed
Characteristics of the Normal
Distribution
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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!
The Shape of the Normal and σ
=5
 = 10
 = 20
0
10
20
30
40
50

60
70
80
90
X
Standard Normal Distribution
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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
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Table of standard normal distribution
PDF template in Excel
Example: X normally distributed with  =
20 and  = 5
Find:
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Probability
Probability
Probability
Probability
that
that
that
that
x
x
x
x
is
is
is
is
more than 30
at least 15
between 15 and 25
between 10 and 30
Percentages of the Area Under a
Normal Curve
Z statistics
1.00
1.96
2.00
2.58
3.00
Show this!
Range





% of the Area
68.26%
95.00%
95.44%
99.00%
99.74%
Percentages of the Area Under a
Normal Curve
68.3%
95.5%
99.7%
-5
-4
-3
-2
-1
0
1
2
3
4
Z score
Example 1. Normal Probability
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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.
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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
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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
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Plotting a Normal Curve
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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
Normal Curve Output
0.03
C2
0.02
0.01
0.00
10
20
30
40
50
60
C1
70
80
90
100
Poisson Process
rate

x x
x
0
time
Assumptions
time homogeneity
independence
no clumping
Poisson Process
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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
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What is the probability that 3
earthquakes will strike during the next
six months?
Poisson Distribution
Count in time period t
e ( t )
P(Y  y ) 
, y  0, 1, 
y!
 t
y
Minitab Probability Calculation
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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
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Duration: Exponential Distribution
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Time between occurrences in a Poisson
process
Continuous probability distribution
Mean =1/t
Exponential Probability Problem
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What is the probability that 9 months
will pass with no earthquake?
t = 1/12 = 1/3
1/ t = 3
Minitab Probability Calculation
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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
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Exponential Probability Density
Function
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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
Exponential Probability Density
Function
0.3
C2
0.2
0.1
0.0
0
5
10
C1
Next Time:
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Random Sampling and Sampling
Distributions
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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