Chapter 4 Continuous Random Variables and Probability Distributions  4.1 - Probability Density Functions.2 - Cumulative Distribution Functions and E Expected.

Slides:



Advertisements
Similar presentations
Chapter 6 Continuous Random Variables and Probability Distributions
Advertisements

Exponential Distribution. = mean interval between consequent events = rate = mean number of counts in the unit interval > 0 X = distance between events.
Hydrologic Statistics
Continuous Distributions
Use of moment generating functions. Definition Let X denote a random variable with probability density function f(x) if continuous (probability mass function.
Ch 4 & 5 Important Ideas Sampling Theory. Density Integration for Probability (Ch 4 Sec 1-2) Integral over (a,b) of density is P(a
Engineering Probability and Statistics - SE-205 -Chap 4 By S. O. Duffuaa.
Probability Densities
Simulation Modeling and Analysis
Continuous Random Variables Chap. 12. COMP 5340/6340 Continuous Random Variables2 Preamble Continuous probability distribution are not related to specific.
Probability and Statistics Review
Some standard univariate probability distributions
A random variable that has the following pmf is said to be a binomial random variable with parameters n, p The Binomial random variable.
Some standard univariate probability distributions
Continuous Random Variables and Probability Distributions
Introduction Before… Next…
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 4 Continuous Random Variables and Probability Distributions.
Some standard univariate probability distributions
Chapter 21 Random Variables Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Exponential, Gamma, Normal Expectation & Variance, Joint.
Some Continuous Probability Distributions Asmaa Yaseen.
4-1 Continuous Random Variables 4-2 Probability Distributions and Probability Density Functions Figure 4-1 Density function of a loading on a long,
Special Continuous Probability Distributions Leadership in Engineering
Important facts. Review Reading pages: P330-P337 (6 th ), or P (7 th )
Standard Statistical Distributions Most elementary statistical books provide a survey of commonly used statistical distributions. The reason we study these.
Functions of Random Variables. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating.
Topic 4 - Continuous distributions
DATA ANALYSIS Module Code: CA660 Lecture Block 3.
Chapter 4 Continuous Random Variables and their Probability Distributions The Theoretical Continuous Distributions starring The Rectangular The Normal.
Chapter 5 Statistical Models in Simulation
Chapter 3 Basic Concepts in Statistics and Probability
Random Variables & Probability Distributions Outcomes of experiments are, in part, random E.g. Let X 7 be the gender of the 7 th randomly selected student.
Statistics for Engineer Week II and Week III: Random Variables and Probability Distribution.
Moment Generating Functions
Some standard univariate probability distributions Characteristic function, moment generating function, cumulant generating functions Discrete distribution.
Functions of Random Variables. Methods for determining the distribution of functions of Random Variables 1.Distribution function method 2.Moment generating.
Use of moment generating functions 1.Using the moment generating functions of X, Y, Z, …determine the moment generating function of W = h(X, Y, Z, …).
Continuous Distributions The Uniform distribution from a to b.
ENGR 610 Applied Statistics Fall Week 3 Marshall University CITE Jack Smith.
CHAPTER Discrete Models  G eneral distributions  C lassical: Binomial, Poisson, etc Continuous Models  G eneral distributions 
More Continuous Distributions
1 Special Continuous Probability Distributions Gamma Distribution Beta Distribution Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS.
1 Lecture 13: Other Distributions: Weibull, Lognormal, Beta; Probability Plots Devore, Ch. 4.5 – 4.6.
The final exam solutions. Part I, #1, Central limit theorem Let X1,X2, …, Xn be a sequence of i.i.d. random variables each having mean μ and variance.
Exam 2: Rules Section 2.1 Bring a cheat sheet. One page 2 sides. Bring a calculator. Bring your book to use the tables in the back.
Ver Chapter 5 Continuous Random Variables 1 Probability/Ch5.
Chapter 5: The Basic Concepts of Statistics. 5.1 Population and Sample Definition 5.1 A population consists of the totality of the observations with which.
Chapter 4 Continuous Random Variables and Probability Distributions  Probability Density Functions.2 - Cumulative Distribution Functions and E Expected.
Chap 5-1 Chapter 5 Discrete Random Variables and Probability Distributions Statistics for Business and Economics 6 th Edition.
Copyright © Cengage Learning. All rights reserved. 4 Continuous Random Variables and Probability Distributions.
DATA ANALYSIS AND MODEL BUILDING LECTURE 4 Prof. Roland Craigwell Department of Economics University of the West Indies Cave Hill Campus and Rebecca Gookool.
Week 61 Poisson Processes Model for times of occurrences (“arrivals”) of rare phenomena where λ – average number of arrivals per time period. X – number.
Theoretical distributions: the other distributions.
Chapter 4 Continuous Random Variables and Probability Distributions
Engineering Probability and Statistics - SE-205 -Chap 4
ASV Chapters 1 - Sample Spaces and Probabilities
The Exponential and Gamma Distributions
Chapter 4 Continuous Random Variables and Probability Distributions
Math 4030 – 10b Inferences Concerning Variances: Hypothesis Testing
Discrete random variable X Examples: shoe size, dosage (mg), # cells,…
Sample Mean Distributions
Chapter 7: Sampling Distributions
Chapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions
Classical Continuous Probability Distributions
Hydrologic Statistics
Distributions and Densities: Gamma-Family and Beta Distributions
Continuous Probability Distributions Part 2
Continuous Probability Distributions Part 2
Continuous Probability Distributions Part 2
Continuous Probability Distributions Part 2
Presentation transcript:

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…