Lecture 9. Continuous Probability Distributions

David R. Merrell90-786 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)

1/8

2/83/84/8

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!

The Shape of the Normal and σ

0 10 20 30 40 50 60 70 80 90

= 5

= 10

= 20

X

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

Percentages of the Area Under a Normal Curve

Z statistics Range % of the Area

1.00 68.26%1.96 95.00%2.00 95.44%2.58 99.00%3.00 99.74%

Show this!

Percentages of the Area Under a Normal Curve

-5 -4 -3 -2 -1 0 1 2 3 4

Z score

68.3%

95.5%

99.7%

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 FunctionNormal 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: Y c2 > X c1 Click: Display > Connect > OK

Normal Curve Output

100908070605040302010

0.03

0.02

0.01

0.00

C1

C2

Poisson Process

time homogeneityindependenceno clumping

rate xxx

0 time

Assumptions

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

P Y ye t

yy

t y

( )( )

!, , ,

0 1

Minitab Probability Calculation

Click: Calc > Probability Distributions > Poisson

Enter: For mean 2, input constant 3 Output:Probability Density FunctionPoisson 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 FunctionExponential 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: Y c2 > X c1 Click: Display > Connect > OK

Exponential Probability Density Function

1050

0.3

0.2

0.1

0.0

C1

C2

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