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Part VI: Named Continuous Random Variables. http://www-users.york.ac.uk/~pml1/bayes/cartoons/cartoon08.jpg. Comparison of Named Distributions. Chapter 30: Continuous Uniform R.V. http://www.six-sigma-material.com/Uniform-Distribution.html. Uniform distribution: Summary.

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Part VI: Named Continuous Random Variables

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### Part VI: Named Continuous Random Variables

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

### Chapter 30: Continuous Uniform R.V.

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

### Uniform distribution: Summary

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

Variable:

X = an exact position or arrival time

Parameter:

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

Density:CDF:

### 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?

### 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?

### 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?

### Chapter 31: Exponential R.V.

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

### 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:

### 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?

### 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?

### 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.)

### Minimum of Two (or More) Exponential Random Variables

Theorem 31.5

If 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.

### Chapter 37: Normal R.V.

http://delfe.tumblr.com/

### Normal Distribution: Summary

Things to look for: bell curve,

Variable:

X = the event

Parameters:

X = the mean

Density:

### PDF of Normal Distribution (cont)

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

### PDF of Normal Distribution

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

### PDF of Normal Distribution (cont)

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

### 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.

### 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

• less 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?

### 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%.

### Chapter 36: Central Limit Theorem(Normal Approximations to Discrete Distributions)

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

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

/normal.html

### Continuity Correction - 1

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

### Continuity Correction - 2

W~N(10, 5)

X ~ Binomial(20, 0.5)

### 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.

### 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

### Gamma Distribution

• Generalization of the exponential function

• Uses

• probability theory

• theoretical statistics

• actuarial science

• operations research

• engineering

### Gamma Function

(t + 1) = t (t), t > 0, t real

(n + 1) = n!, n > 0, n integer

### Gamma Distribution: Summary

Things to look for: waiting time until rth event occurs

Variable: X = time until the rth event occurs, X ≥ 0

Parameters:

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

: the average rate

Density:

### Gamma Random Variable

k = r

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

### 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

### 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)

X

### Other Continuous Random Variables

• Weibull

• exponential is a member of family

• lognormal

• log of the normal distribution

• uses: products of distributions

• Cauchy

• symmetrical, flatter than normal

### 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

### Example: Determine the Distribution (class)

For each of the following situations, state which distribution would be appropriate and why. Also please state all parameters. Note: Since we did not discuss the Gamma and Beta, those should be answered as ‘none’. In the final, you will need to know the appropriate approximations also.

Exercises: 37.1 – 37.12 (pp. 522 – 523)

Answer to 37.11 is wrong in the back of the book.