Part VI: Named Continuous Random Variables - PowerPoint Presentation

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Part VI: Named Continuous Random Variables

http://www-users.york.ac.uk/~pml1/bayes/cartoons/cartoon08.jpg

1Slide2

Comparison of Named Distributions

discrete

continuous

Bernoulli,

Binomial, Geometric, Negative Binomial, Poisson, Hypergeometric, Discrete Uniform

Continuous Uniform, Exponential, Gamma, Beta, Normal

2Slide3

Chapter 30: Continuous Uniform R.V.

http://www.six-sigma-material.com/Uniform-Distribution.html

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Uniform distribution: Summary

Things to look for: constant density on a line or area

Variable: X = an exact position or arrival timeParameter:

(

a,b): the endpoints where the density is nonzero.Density: CDF:

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Example: Uniform Distribution (Class)

A bus arrives punctually at a bus stop every thirty minutes. Each morning, a bus rider leaves her house and casually strolls to the bus stop.

Why is this a Continuous Uniform distribution situation? What are the parameters? What is X?What is the density for the wait time in minutes?

What is the CDF for the wait time in minutes?

Graph the density.

Graph the CDF.What is the expected wait time?5Slide6

Example: Uniform Distribution (Class)

A bus arrives punctually at a bus stop every thirty minutes. Each morning, a bus rider leaves her house and casually strolls to the bus stop.

What is the standard deviation for the wait time?What is the probability that the person will wait between 20 and 40 minutes? (Do this via 3 different methods.)

Given that the person waits at least 15 minutes, what is the probability that the person will wait at least 20 minutes?

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Example: Uniform Distribution

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Example: Uniform Distribution (Class)

A bus arrives punctually at a bus stop every thirty minutes. Each morning, a bus rider leaves her house and casually strolls to the bus stop.

Let the cost of this waiting be $20 per minute plus an additional $5.What are the parameters?

What is the density for the cost in minutes?

What is the CDF for the cost in minutes?

What is the expected cost to the rider?What is the standard deviation of the cost to the rider?8Slide9

Chapter 31: Exponential R.V.

http://en.wikipedia.org/wiki/Exponential_distribution

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Exponential Distribution: Summary

Things to look for: waiting time until first event occurs or time between events.

Variable:

X = time until the next event occurs, X ≥ 0

Parameter:

: the average rateDensity: CDF:

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Example: Exponential R.V. (class)

Suppose that the arrival time (on average) of a large earthquake in Tokyo occurs with an exponential distribution with an average of 8.25 years.

What does X represent in this story? What values can X take?Why is this an example of the Exponential distribution?

What is the parameter for this distribution?

What is the density?

What is the CDF?What is the standard deviation for the next earthquake?11Slide12

Example: Exponential R.V. (class, cont.)

Suppose that the arrival time (on average) of a large earthquake in Tokyo occurs with an exponential distribution with an average of 8.25 years.

What is the probability that the next earthquake occurs after three but before eight years?What is the probability that the next earthquake occurs before 15 years

?

What is the probability that the next earthquake occurs after 10 years

?How long would you have to wait until there is a 95% chance that the next earthquake will happen?12Slide13

Example: Exponential R.V. (Class, cont.)

Suppose that the arrival time (on average) of a large earthquake in Tokyo occurs with an exponential distribution with an average of 8.25 years.

k) Given that there has been no large Earthquakes in Tokyo for more than 5 years, what is the chance that there will be a large Earthquake in Tokyo in more than 15 years? (Do this problem using the memoryless

property and the definition of conditional probabilities.)

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Minimum of Two (or More) Exponential Random Variables

Theorem 31.5If X1

, …, Xn are independent exponential random variables with parameters 

1

, …,

n then Z =  min(X1, …, Xn) is an exponential random variable with parameter 1 + … + n.14Slide15

Chapter 37: Normal R.V.

http://delfe.tumblr.com/

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Normal Distribution: Summary

Things to look for: bell curve,

Variable: X = the eventParameters:



X = the meanDensity:

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PDF of Normal Distribution (cont)

http://commons.wikimedia.org/wiki/File:Normal_distribution_pdf.svg

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PDF of Normal Distribution

http://www.oswego.edu/~srp/stats/z.htm

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PDF of Normal Distribution (cont)

http://commons.wikimedia.org/wiki/File:Normal_distribution_pdf.svg

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Procedure for doing Normal Calculations

Sketch the problem.

Write down the probability of interest in terms of the original problem.Convert to standard normal.

Convert to CDFs.

Use the z-table to write down the values of the CDFs.

Calculate the answer.21Slide22

Example: Normal r.v. (Class)

The gestation periods of women are normally distributed with

 = 266 days and  = 16 days. Determine the probability that a gestation period is

l

ess than 225 days.

between 265 and 295 days.more than 276 days.less than 300 days.Among women with a longer than average gestation, what is the probability that they give birth longer than 300 days?22Slide23

Example: “Backwards” Normal r.v. (Class)

The gestation periods of women are normally distributed with

 = 266 days and  = 16 days. Find the gestation length for the following situations:

longest

6%.

shortest 13%.middle 50%.23Slide24

Example: Normal r.v. (class)

Suppose that a Scottish soldier's chest size is normally distributed with

 = 39.8 inches and  = 2.05 inches. a) What is the probability that a Scottish soldier has a chest size of less than 35 inches?

b

) What is the probability that a Scottish soldier has a chest of at least 40 inches?

c) What is the probability that a Scottish soldier has a chest size between 35 and 40 inches?24Slide25

Chapter 35: Sums of Independent Normal Random Variables

https://www.statsoft.com/Textbook/Cluster-Analysis

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Chapter 36: Central Limit Theorem(Normal Approximations to Discrete Distributions – 36.4, 36.5)

http://nestor.coventry.ac.uk/~nhunt/binomial/normal.html

http://nestor.coventry.ac.uk/~

nhunt/poisson

/

normal.html26Slide27

Continuity Correction - 1

http://www.marin.edu/~npsomas/Normal_Binomial.htm

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Continuity Correction - 2

X

~ Binomial(20, 0.5)

W~

N

(10, 5)28Slide29

Continuity Correction - 3

Discrete

Continuous

a < X

a

+ 0.5 < Xa ≤ Xa – 0.5 < X X < b X < b – 0.5 X ≤ b X < b + 0.529Slide30

Normal Approximation to Binomial

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Example: Normal Approximation to Binomial (Class)

The ideal size of a first-year class at a particular college is 150 students. The college, knowing from past experience that on the average only 30 percent of these accepted for admission will actually attend, uses a policy of approving the applications of 450 students.

Compute the probability that more than 150 students attend this college.Compute the probability that fewer than 130 students attend this college.

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Chapter 32: Gamma R.V.

http://

resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref

/process_simulations_sensitivity_analysis_and_error_analysis_modeling

/

distributions_for_assigning_random_values.htm32Slide33

Gamma Distribution

Generalization of the exponential functionUsesprobability theorytheoretical statistics

actuarial scienceoperations researchengineering

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Gamma Function

(t + 1) = t (t), t > 0, t real

(n + 1) = n!, n > 0, n integer

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Gamma Distribution: Summary

Things to look for: waiting time until

rth event occurs

Variable: X = time until the

rth

event occurs, X ≥ 0Parameters: r: total number of arrivals/events that you are waiting for : the average rateDensity:

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Gamma Random Variable

http://en.wikipedia.org/wiki/File:Gamma_distribution_pdf.svg

k = r

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Chapter 33: Beta R.V.

http://mathworld.wolfram.com/BetaDistribution.html

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Beta Distribution

This distribution is only defined on an intervalstandard beta is on the interval [0,1]The formula in the book is for the standard beta

usesmodeling proportionspercentagesprobabilities

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Beta Distribution: Summary

Things to look for: percentage, proportion, probability

Variable: X = percentage, proportion, probability of interest (standard Beta)

Parameters:

, Density:

Density: no simple form

When A = 0, B = 1 (Standard Beta)

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Shapes of Beta Distribution

http://upload.wikimedia.org/wikipedia/commons/9/9a/Beta_distribution_pdf.png

X

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Other Continuous Random Variables

Weibullexponential is a member of family

uses: lifetimeslognormallog of the normal distributionuses: products of distributions

Cauchy

symmetrical, flatter than normal

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Chapter 37: Summary and Review of Named Continuous R.V.

http://

www.wolfram.com/mathematica/new-in-8/parametric-probability-distributions

/

univariate-continuous-distributions.html

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Summary of Continuous Distributions

Name

Density,

f

X

(x)DomainCDF, FX(x)(X)Var(X)ParametersWhat X isWhen used

Name

Density,

f

X

(x)

Domain

CDF, FX(x)Var(X)ParametersWhat X isWhen used43Slide44

Expected values and Variances for selected families of continuous random variables.

Family

Parameter(s)

Expectation

Variance

Uniforma,bExponentiallNormalm,s2m2

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