Part VI: Named Continuous Random Variables http://www-users.york.ac.uk/~pml1/bayes/cartoons/cartoon08.jpg

Comparison of Named Distributions discrete continuous Bernoulli, Binomial, Geometric, Negative Binomial, Poisson, Hypergeometric, Discrete Uniform Continuous Uniform, Exponential, Gamma, Beta, Normal

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: 𝑓 𝑥 𝑥 = 1 𝑏−𝑎 𝑎≤𝑥≤𝑏 0 𝑒𝑙𝑠𝑒 𝐹 𝑋 𝑥 = 0 𝑥<𝑎 𝑥−𝑎 𝑏−𝑎 𝑎≤𝑥≤𝑏 1 𝑏<𝑥 𝔼 𝑋 = 𝑎+𝑏 2 , 𝑉𝑎𝑟 𝑋 = (𝑏−𝑎) 2 12

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

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: 𝑓 𝑥 𝑥 = 𝜆 𝑒 −𝜆𝑥 𝑥>0 0 𝑒𝑙𝑠𝑒 𝐹 𝑋 𝑥 = 1− 𝑒 −𝜆𝑥 𝑥>0 0 𝑒𝑙𝑠𝑒 𝔼 𝑋 = 1 𝜆 , 𝑉𝑎𝑟 𝑋 = 1 𝜆 2

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 𝜎 𝑋 2 =𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Density: 𝑓 𝑥 𝑥 = 𝑒 − (𝑥− 𝜇 𝑥 ) 2 (2 𝜎 2 ) 2𝜋 𝜎 2 , 𝑥 𝜖 ℝ 𝔼 𝑋 =X, 𝑉𝑎𝑟 𝑋 = 𝜎 𝑋 2

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. Calculate the answer.

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%.

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?

Chapter 35: Sums of Independent Normal Random Variables https://www.statsoft.com/Textbook/Cluster-Analysis

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

Continuity Correction - 1 http://www.marin.edu/~npsomas/Normal_Binomial.htm

Continuity Correction - 2 W~N(10, 5) X ~ Binomial(20, 0.5)

Continuity Correction - 3 Discrete Continuous a < X a + 0.5 < X a ≤ X a – 0.5 < X X < b X < b – 0.5 X ≤ b X < b + 0.5

Normal Approximation to Binomial

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: 𝑓 𝑥 𝑥 = 𝜆 𝑟 Γ(𝑟) 𝑥 𝑟−1 𝑒 −𝜆𝑥 𝑥>0 0 𝑒𝑙𝑠𝑒 𝔼 𝑋 = 𝑟 𝜆 , 𝑉𝑎𝑟 𝑋 = 𝑟 𝜆 2 𝐶𝐷𝐹: 𝐹 𝑋 𝑥 = 1− 𝑒 −𝜆𝑥 𝑗=0 𝑟−1 (𝜆𝑥) 𝑗 𝑗! 𝑥>0 0 𝑒𝑙𝑠𝑒

Gamma Random Variable k = r 𝜃= 1 𝜆 http://en.wikipedia.org/wiki/File:Gamma_distribution_pdf.svg

Chapter 33: Beta R.V. http://mathworld.wolfram.com/BetaDistribution.html

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: 𝑓 𝑥 𝑥 = 1 𝐵−𝐴 Γ(𝛼+𝛽) Γ(𝛼)Γ(𝛽) 𝑥−𝐴 𝐵−𝐴 𝛼−1 𝑥−𝐴 𝐵−𝐴 𝛽−1 𝐴≤𝑥≤𝐵 0 𝑒𝑙𝑠𝑒 Density: no simple form When A = 0, B = 1 (Standard Beta) 𝔼 𝑋 = 𝛼 𝛼+𝛽 , 𝑉𝑎𝑟 𝑋 = 𝛼𝛽 𝛼+𝛽 2 (𝛼+𝛽+1)

Shapes of Beta Distribution http://upload.wikimedia.org/wikipedia/commons/9/9a/Beta_distribution_pdf.png X

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

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

Summary of Continuous Distributions Name Density, fX(x) Domain CDF, FX(x) 𝔼(X) Var(X) Parameters What X is When used

Expected values and Variances for selected families of continuous random variables. Family Parameter(s) Expectation Variance Uniform a,b Exponential l Normal m,s2 m 2