ASV Chapters 1 - Sample Spaces and Probabilities 2 - Conditional Probability and Independence 3 - Random Variables 4 - Approximations of the Binomial Distribution 5 - Transforms and Transformations 6 - Joint Distribution of Random Variables 7 - Sums and Symmetry 8 - Expectation and Variance in the Multivariate Setting 9 - Tail Bounds and Limit Theorems 10 - Conditional Distribution 11 - Appendix A, B, C, D, E, F

Poisson Distribution (discrete) For x = 0, 1, 2, …, this calculates P(x Events) in a random sample of n trials coming from a population with rare P(Event) = . But it may also be used to calculate P(x Events) within a random interval of time units, for a “Poisson process” having a known “Poisson rate” α. Recall… T X = # “clicks” on a Geiger counter in normal background radiation.

Poisson Distribution (discrete) For x = 0, 1, 2, …, this calculates P(x Events) in a random sample of n trials coming from a population with rare P(Event) = . But it may also be used to calculate P(x Events) within a random interval of time units, for a “Poisson process” having a known “Poisson rate” α. T X = time between “clicks” on a Geiger counter in normal background radiation. X = # “clicks” on a Geiger counter in normal background radiation. “Time-to-Event Analysis” “Time-to-Failure Analysis” “Reliability Analysis” “Survival Analysis” failures, deaths, births, etc. Time between events is often modeled by the Exponential Distribution (continuous).

Time between events is often modeled by the Exponential Distribution (continuous). X ~ Exp() parameter  > 0 Check pdf?   X = Time between events

Time between events is often modeled by the Exponential Distribution (continuous). X ~ Exp() parameter  > 0 Calculate the expected time between events X = Time between events

Time between events is often modeled by the Exponential Distribution (continuous). X ~ Exp() parameter  > 0 Calculate the expected time between events Similarly for the variance… etc... = X = Time between events

Time between events is often modeled by the Exponential Distribution (continuous). X ~ Exp() parameter  > 0 Calculate the expected time between events Determine the cdf X = Time between events

Calculate the expected time between events Time between events is often modeled by the Exponential Distribution (continuous). X ~ Exp() parameter  > 0 Calculate the expected time between events Determine the cdf Note: “Reliability Function” R(t) “Survival Function” S(t) X = Time between events

Time between events is often modeled by the Exponential Distribution (continuous). X ~ Exp() parameter  > 0 Example: Suppose mean time between events is known to be… = 2 years Then for x  0, Calculate Calculate the “Poisson rate” . X = Time between events

Poisson Distribution (discrete) For x = 0, 1, 2, …, this calculates P(x Events) in a random sample of n trials coming from a population with rare P(Event) = . But it may also be used to calculate P(x Events) within a random interval of time units, for a “Poisson process” having a known “Poisson rate” α. T The mean number of events during this time interval (0, T) is . Therefore, the mean number of events in one unit of time is . X = Time between events is often modeled by the Exponential Distribution (continuous). Connection? However, the mean time between events was just shown to be = . Ex: Suppose the mean number of instantaneous clicks/sec is  = 10, then the mean time between any two successive clicks is  = 1/10 sec. Ex: Suppose the mean number of instantaneous clicks/sec is  = 10, then the mean time between any two successive clicks is  = 1/10 sec. 1 second

Time between events is often modeled by the Exponential Distribution (continuous). X ~ Exp() parameter  > 0 Example: Suppose mean time between events is known to be… = 2 years Then for x  0, Calculate Calculate the “Poisson rate” . X = Time between events

independent of time t; only depends on t Another property … (Event = “Failure,” etc.) T No Failure What is the probability of “No Failure” up to t +  t, given “No Failure” up to t? independent of time t; only depends on t “Memory-less” property of the Exponential distribution The conditional property of “no failure” from ANY time t to a future time t + t of fixed duration t, remains constant. Models many systems in the “prime of their lives,” e.g., a random 30-yr old individual in the USA.

The Gamma Distribution More general models exist…, e.g., The Gamma Distribution In order to understand this, it is first necessary to understand the ”Gamma Function” Def: For any  > 0, Discovered by Swiss mathematician Leonhard Euler (1707-1783) in a different form. “Special Functions of Mathematical Physics” includes Gamma, Beta, Bessel, classical orthogonal polynomials (Jacobi, Chebyshev, Legendre, Hermite,…), etc. Generalization of “factorials” to all complex values of  (except 0, -1, -2, -3, …). The Exponential distribution is a special case of the Gamma distribution! Basic Properties:  Proof: Proof: Let  = n = 1, 2, 3, … 

The Gamma Function

 General Gamma Distribution Gamma Function  = “shape parameter”  = “scale parameter” Exponential Distribution Note that if  = 1, then pdf Note that if  = 1, then pdf  Standard Gamma Distribution

General Gamma Distribution Standard Gamma Distribution WLOG… General Gamma Distribution Gamma Function  = “shape parameter” Standard Gamma Distribution

Standard Gamma Distribution General Gamma Distribution WLOG… Standard Gamma Distribution General Gamma Distribution Gamma Function  = “shape parameter”

Standard Gamma Distribution Gamma Function  = “shape parameter”

Standard Gamma Distribution “Incomplete Gamma Function”  = “shape parameter” “Incomplete Gamma Function” (No general closed form expression, but still continuous and monotonic from 0 to 1.)

General Gamma Distribution Exponential Distribution Return to… Gamma Function  = “shape parameter”  = “scale parameter” Exponential Distribution Note that if  = 1, then “Poisson rate”  = 1/ =  “independent, identically distributed” (i.i.d.) Theorem: Suppose r.v.’s Then their sum e.g., failure time in machine components

General Gamma Distribution Gamma Function  = “shape parameter”  = “scale parameter” Example: Suppose X = time between failures is known to be modeled by a Gamma distribution, with mean = 8 years, and standard deviation = 4 years. Calculate the probability of failure before 5 years.

General Gamma Distribution Gamma Function  = “shape parameter”  = “scale parameter” Example: Suppose X = time between failures is known to be modeled by a Gamma distribution, with mean = 8 years, and standard deviation = 4 years. Calculate the probability of failure before 5 years. 5.6 3

Chi-Squared Distribution with  = n  1 degrees of freedom df = 1, 2, 3,…  = 1  = 2  = 3  = 4  = 5  = 6  = 7 Special case of the Gamma distribution: “Chi-squared Test” used in statistical analysis of categorical data.

with degrees of freedom 1 and 2 . F-distribution with degrees of freedom 1 and 2 . “F-Test” used when comparing means of two or more groups (ANOVA).

with (n – 1) degrees of freedom df = 1, 2, 3, … T-distribution with (n – 1) degrees of freedom df = 1, 2, 3, … df = 1 df = 2 df = 5 df = 10 “T-Test” used when analyzing means of one or two groups.

T-distribution with 1 degree of freedom “Cauchy distribution” df = 1

 T-distribution “Cauchy distribution” with 1 degree of freedom improper integral at both endpoints 

 T-distribution “Cauchy distribution” with 1 degree of freedom improper integral at both endpoints 

“Cauchy distribution” T-distribution with 1 degree of freedom “Cauchy distribution” improper integral at both endpoints “indeterminate form”

 does not exist! “indeterminate form” T-distribution with 1 degree of freedom “Cauchy distribution” improper integral at both endpoints  does not exist! “indeterminate form”

Classical Continuous Probability Distributions Normal distribution Log-Normal ~ X is not normally distributed (e.g., skewed), but Y = “logarithm of X” is normally distributed Student’s t-distribution ~ Similar to normal distr, more flexible F-distribution ~ Used when comparing multiple group means Chi-squared distribution ~ Used extensively in categorical data analysis Others for specialized applications ~ Gamma, Beta, Weibull…