Chapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions § 4. 1 - Probability Density Functions § 4. 2 - Cumulative Distribution Functions and Expected Values § 4. 3 - The Normal Distribution § 4. 4 - The Exponential and Gamma Distributions § 4. 5 - Other Continuous Distributions § 4. 6 - Probability Plots
Recall… 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. 0 X = # “clicks” on a Geiger counter in normal background radiation. T
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. 0 T X = #time “clicks” between on a “clicks” Geiger on counter a in Geiger normalcounter background in normal radiation. background radiation. failures, deaths, births, etc. • “Time-to-Event Analysis” • “Time-to-Failure Analysis” • “Reliability Analysis” • “Survival Analysis” 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 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 Similarly for the variance… X = Time between events etc. . . =
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
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 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. 0 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. . 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 the “Poisson rate” . X = Time between events
Another property … (Event = “Failure, ” etc. ) 0 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.
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: Let = n = 1, 2, 3, … �
The Gamma Function
General Gamma Distribution Gamma Function = “shape parameter” = “scale parameter” Note that if = 1, then pdf Exponential Distribution Note that if = 1, then pdf � Standard Gamma Distribution
WLOG… General Gamma Distribution Gamma Function = “shape parameter” Standard Gamma Distribution
WLOG… Standard General Gamma Distribution Gamma Function = “shape parameter”
Standard Gamma Distribution Gamma Function = “shape parameter”
Standard Gamma Distribution Gamma Function = “shape parameter” “Incomplete Gamma Function” (No general closed form expression, but still continuous and monotonic from 0 to 1. )
Return to… General Gamma Distribution Gamma Function = “shape parameter” = “scale parameter” Note that if = 1, then Exponential Distribution “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 5. 68 years, and standard deviation = 3 mean = 4 years. Calculate the probability of failure before 5 years.
Chi-Squared Distribution with = n 1 degrees of freedom df = 1, 2, 3, … Special case of the Gamma distribution: =1 =2 =3 =4 =5 =6 “Chi-squared Test” used in statistical analysis of categorical data. =7 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, … df = 1 df = 2 df = 5 df = 10 “T-Test” used when analyzing means of one or two groups. 25
T-distribution “Cauchy distribution” with 1 degree of freedom df = 1 26
T-distribution “Cauchy distribution” with 1 degree of freedom improper integral at both endpoints � 27
T-distribution “Cauchy distribution” with 1 degree of freedom improper integral at both endpoints � 28
T-distribution “Cauchy distribution” with 1 degree of freedom improper integral at both endpoints “indeterminate form” 29
T-distribution “Cauchy distribution” with 1 degree of freedom does not exist! improper integral at both endpoints “indeterminate form” 30
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… 31
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