Chapter 4 Continuous Random Variables and Probability Distributions  Probability Density Functions.2 - Cumulative Distribution Functions and E Expected Values  The Normal Distribution.4 - The Exponential and Gamma Distributions.5 - Other Continuous Distributions.6 - Probability Plots

X = # “clicks” on a Geiger counter in normal background radiation. 0T 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. 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.

X = # “clicks” on a Geiger counter in normal background radiation. 0T X = time between “clicks” on a Geiger counter in normal background radiation. Exponential Distribution (continuous) Time between events is often modeled by the Exponential Distribution (continuous). failures, deaths, births, etc. “Time-to-Event Analysis” “Time-to-Failure Analysis” “Reliability Analysis” “Survival Analysis” 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. 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.

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

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

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

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

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

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

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. 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. Ex: Suppose the mean number of instantaneous clicks/sec is  = 10, then the mean time between any two successive clicks is  = 1/10 sec. 0T Exponential Distribution (continuous) X = Time between events is often modeled by the Exponential Distribution (continuous). mean number of events The mean number of events during this time interval (0, T) is. mean number of eventsone unit Therefore, the mean number of events in one unit of time is. mean time between events However, the mean time between events was just shown to be =.Connection? 1 second

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

0T Another property … (Event = “Failure,” etc.) 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” “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.

More general models exist…, e.g., ”Gamma Function” In order to understand this, it is first necessary to understand the ”Gamma Function” Def: Def: For any  > 0, Discovered by Swiss mathematician Leonhard Euler ( ) 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:   Let  = n = 1, 2, 3, …

The Gamma Function

 Gamma Function Note that if  = 1, then pdf is Note that if  = 1, then pdf is  = “shape parameter”  = “scale parameter”

Gamma Function  = “shape parameter”  = “scale parameter”  = 1 WLOG…

Gamma Function  = “shape parameter”  = “scale parameter”

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

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

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.

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

22 = 1 = 2 = 3 = 4 = 5 = 6 = 7 Chi-Squared Distribution with = n  1 degrees of freedom df = 1, 2, 3,… Special case of the Gamma distribution: “Chi-squared Test” used in statistical analysis of categorical data.

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

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

25 T-distribution with 1 degree of freedom df = 1

26 T-distribution with 1 degree of freedom improper integral at both endpoints 

 27 T-distribution with 1 degree of freedom improper integral at both endpoints

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

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

30 ● 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…