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Part VI: Named Continuous Random Variables

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

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Comparison of Named Distributionsdiscrete continuous

Bernoulli, Binomial, Geometric, Negative

Binomial, Poisson, Hypergeometric, Discrete Uniform

Continuous Uniform, Exponential, Gamma,

Beta, Normal

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Chapter 30: Continuous Uniform R.V.

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

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Uniform distribution: SummaryThings to look for: constant density on a line or areaVariable:

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.

a) Why is this a Continuous Uniform distribution situation? What are the parameters? What is X?

b) What is the density for the wait time in minutes?c) What is the CDF for the wait time in minutes?d) Graph the density.e) Graph the CDF.f) What is the expected wait time?

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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.

g) What is the standard deviation for the wait time?

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

i) 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

-10 -5 0 5 10 15 20 25 30 35 400.00

0.01

0.02

0.03

0.04

-10 -5 0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

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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.

a) What are the parameters?b) What is the density for the cost in minutes?c) What is the CDF for the cost in minutes?d) What is the expected cost to the rider?e) What is the standard deviation of the cost to the

rider?

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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 rate

Density: 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.

a) What does X represent in this story? What values can X take?

b) Why is this an example of the Exponential distribution?

c) What is the parameter for this distribution?d) What is the density?e) What is the CDF?f) What is the standard deviation for the next

earthquake?

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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.

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

h) What is the probability that the next earthquake occurs before 15 years?

i) What is the probability that the next earthquake occurs after 10 years?

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

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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.

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Chapter 37: Normal R.V.

http://delfe.tumblr.com/

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Normal Distribution: SummaryThings to look for: bell curve,Variable:

X = the eventParameters:

X = the mean

Density:

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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

X 00.2

X ( 2)0.5

X 05

X 01

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Procedure for doing Normal Calculations

1) Sketch the problem.2) Write down the probability of interest in

terms of the original problem.3) Convert to standard normal.4) Convert to CDFs.5) Use the z-table to write down the values of

the CDFs.6) Calculate the answer.

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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

a) less than 225 days.b) between 265 and 295 days.c) more than 276 days.d) less than 300 days.e) Among women with a longer than average

gestation, what is the probability that they give birth longer than 300 days?

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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:

a) longest 6%.b) shortest 13%.c) middle 50%.

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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.html

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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)

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Continuity Correction - 3Discrete Continuousa < X a + 0.5 < Xa ≤ X a – 0.5 < X

X < b X < b – 0.5 X ≤ b X < b + 0.5

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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.

a) Compute the probability that more than 150 students attend this college.

b) 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.htm

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Gamma Distribution

• Generalization of the exponential function• Uses–probability theory– theoretical statistics– actuarial science–operations research–engineering

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Gamma Function

(t + 1) = t (t), t > 0, t real(n + 1) = n!, n > 0, n integer

t 1 x

0(t) x e dx,t 0

2n

1 (2n)!n

2 n!2

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Gamma Distribution: SummaryThings to look for: waiting time until rth event occursVariable: X = time until the rth event occurs, X ≥ 0Parameters:

r: total number of arrivals/events that you are waiting for: the average rate

Density:

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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 interval– standard beta is on the interval [0,1]– The formula in the book is for the standard

beta• uses–modeling proportions–percentages–probabilities

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Beta Distribution: SummaryThings to look for: percentage, proportion, probabilityVariable: X = percentage, proportion, probability of interest

(standard Beta)Parameters:

, Density:

Density: no simple formWhen 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

• Weibull–exponential is a member of family–uses: lifetimes

• lognormal– log of the normal distribution–uses: 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 DistributionsName

Density, fX(x)

DomainCDF, FX(x)

(X)Var(X)Parameters

What X is

When used

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

Family Parameter(s) Expectation Variance

Uniform a,b

Exponential

Normal ,2 2

1 2

1

a b2 2(b a)

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