continuous distributions
play

Continuous Distributions 1.8-1.9: Continuous Random Variables - PowerPoint PPT Presentation

Jun 18, 2023 •157 likes •489 views

Continuous Distributions 1.8-1.9: Continuous Random Variables 1.10.1: Uniform Distribution (Continuous) 1.10.4-5 Exponential and Gamma Distributions: Distance between crossovers Prof. Tesler Math 283 Fall 2019 Prof. Tesler Continuous

  1. Continuous Distributions 1.8-1.9: Continuous Random Variables 1.10.1: Uniform Distribution (Continuous) 1.10.4-5 Exponential and Gamma Distributions: Distance between crossovers Prof. Tesler Math 283 Fall 2019 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 1 / 33

  2. Cumulative Distribution Function (CDF) Prof. Tesler Continuous Distributions Math 283 / Fall 2019 2 / 33

  3. Cumulative Distribution Function (CDF) Discrete random variables The Cumulative Distribution Function (CDF) of PDF random variable X is P X ( k ) k F X ( x ) = P ( X � x ) 0 . 5 0 . 1 1 . 0 0 . 2 F X ( 1 . 5 ) = P ( X � 1 . 5 ) = P X ( 0 . 5 ) + P X ( 1 . 0 ) + P X ( 1 . 5 ) 1 . 5 0 . 3 = 0 . 1 + 0 . 2 + 0 . 3 = 0 . 6 2 . 0 0 . 1 2 . 5 0 . 1 3 . 0 0 . 2 In-between points with nonzero probability: F X ( 1 . 7 ) = P ( X � 1 . 7 ) = P ( X � 1 . 5 ) = F X ( 1 . 5 ) = 0 . 6 whereas the PDF there is 0 : P X ( 1 . 7 ) = 0 Similarly, F X ( k ) = F X ( 1 . 5 ) = 0 . 6 for 1 . 5 � k < 2 . 0 . Prof. Tesler Continuous Distributions Math 283 / Fall 2019 3 / 33

  4. CDF outside of the range PDF P X ( k ) k 0 . 5 0 . 1 1 . 0 0 . 2 1 . 5 0 . 3 2 . 0 0 . 1 2 . 5 0 . 1 3 . 0 0 . 2 F X (− 1 ) = P ( X � − 1 ) = 0 (no points w/nonzero PDF) F X ( 5 ) = P ( X � 5 ) = 1 (has all of the points w/nonzero PDF) General case k → − ∞ F X ( k ) = 0 lim k → + ∞ F X ( k ) = 1 lim Prof. Tesler Continuous Distributions Math 283 / Fall 2019 4 / 33

  5. CDF table PDF CDF P X ( k ) F X ( k ) k k k < 0 . 5 0 0 . 5 0 . 1 0 . 5 � k < 1 . 0 0 . 1 1 . 0 0 . 2 1 . 0 � k < 1 . 5 0 . 3 1 . 5 0 . 3 1 . 5 � k < 2 . 0 0 . 6 2 . 0 0 . 1 2 . 0 � k < 2 . 5 0 . 7 2 . 5 0 . 1 2 . 5 � k < 3 . 0 0 . 8 3 . 0 0 . 2 3 . 0 � k 1 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 5 / 33

  6. Using CDF table with various inequalities: � , > , < , � PDF CDF P X ( k ) F X ( k ) k k k < 0 . 5 0 0 . 5 0 . 1 0 . 5 � k < 1 . 0 0 . 1 1 . 0 0 . 2 1 . 0 � k < 1 . 5 0 . 3 1 . 5 0 . 3 1 . 5 � k < 2 . 0 0 . 6 2 . 0 0 . 1 2 . 0 � k < 2 . 5 0 . 7 2 . 5 0 . 1 2 . 5 � k < 3 . 0 0 . 8 3 . 0 0 . 2 3 . 0 � k 1 P ( X � 1 ) = 0 . 3 P ( X > 1 ) = 1 − P ( X � 1 ) = 0 . 7 P ( X < 1 ) = P ( X � 1 − ) = F X ( 1 − ) = 0 . 1 using infinitesimal notation from Calculus: 1 − is just below 1 , like 0 . 99999999 , but even closer. P ( X � 1 ) = 1 − P ( X < 1 ) = 1 − F X ( 1 − ) = 0 . 9 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 6 / 33

  7. Using CDF table on an interval PDF CDF P X ( k ) F X ( k ) k k k < 0 . 5 0 0 . 5 0 . 1 0 . 5 � k < 1 . 0 0 . 1 1 . 0 0 . 2 1 . 0 � k < 1 . 5 0 . 3 1 . 5 0 . 3 1 . 5 � k < 2 . 0 0 . 6 2 . 0 0 . 1 2 . 0 � k < 2 . 5 0 . 7 2 . 5 0 . 1 2 . 5 � k < 3 . 0 0 . 8 3 . 0 0 . 2 3 . 0 � k 1 F X ( 2 ) = P ( X � 2 ) = P X ( 0 . 5 ) + P X ( 1 . 0 ) + P X ( 1 . 5 ) + P X ( 2 . 0 ) F X ( 1 ) = P ( X � 1 ) = P X ( 0 . 5 ) + P X ( 1 . 0 ) P ( 1 < X � 2 ) = P X ( 1 . 5 ) + P X ( 2 . 0 ) = P ( X � 2 ) − P ( X � 1 ) = F X ( 2 ) − F X ( 1 ) = 0 . 7 − 0 . 3 = 0 . 4 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 7 / 33

  8. Converting intervals to the form P ( a < X � b ) PDF CDF P X ( k ) F X ( k ) k k k < 0 . 5 0 0 . 5 0 . 1 0 . 5 � k < 1 . 0 0 . 1 1 . 0 0 . 2 1 . 0 � k < 1 . 5 0 . 3 1 . 5 0 . 3 1 . 5 � k < 2 . 0 0 . 6 2 . 0 0 . 1 2 . 0 � k < 2 . 5 0 . 7 2 . 5 0 . 1 2 . 5 � k < 3 . 0 0 . 8 3 . 0 0 . 2 3 . 0 � k 1 The formula P ( a < X � b ) = F X ( b ) − F X ( a ) uses a < X (not a � X ) and X � b (not X < b ). Other formats must be converted to this: P ( 1 < X � 2 ) = F X ( 2 ) − F X ( 1 ) = 0 . 7 − 0 . 3 = 0 . 4 P ( 1 � X � 2 ) = P ( 1 − < X � 2 ) = F X ( 2 ) − F X ( 1 − ) = 0 . 7 − 0 . 1 = 0 . 6 P ( 1 < X < 2 ) = P ( 1 < X � 2 − ) = F X ( 2 − ) − F X ( 1 ) = 0 . 6 − 0 . 3 = 0 . 3 P ( 1 � X < 2 ) = P ( 1 − < X � 2 − ) = F X ( 2 − ) − F X ( 1 − ) = 0 . 6 − 0 . 1 = 0 . 5 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 8 / 33

  9. Continuous distributions Prof. Tesler Continuous Distributions Math 283 / Fall 2019 9 / 33

  10. Continuous distributions Example Pick a real number x between 20 and 30 with all real values in [ 20 , 30 ] equally likely. Sample space: S = [ 20 , 30 ] Number of outcomes: | S | = ∞ Probability of each outcome: P ( X = x ) = 1 ∞ = 0 Yet, P ( X � 21 . 5 ) = 15 % Prof. Tesler Continuous Distributions Math 283 / Fall 2019 10 / 33

  11. Continuous distributions The sample space S is often a subset of R n . We’ll do the 1-dimensional case S ⊂ R . The probability density function (PDF) f X ( x ) is defined differently than the discrete case: f X ( x ) is a real-valued function on S with f X ( x ) � 0 for all x ∈ S . � (vs. � f X ( x ) dx = 1 P X ( x ) = 1 for discrete) x ∈ S S � (vs. � The probability of event A ⊂ S is P ( A ) = f X ( x ) dx P X ( x ) ). x ∈ A A In n dimensions, use n -dimensional integrals instead. Notation: Uppercase F for CDF vs. lowercase f for pdf. Uniform distribution Let a < b be real numbers. The Uniform Distribution on [ a , b ] is that all numbers in [ a , b ] are “equally likely.” � 1 if a � x � b ; b − a More precisely, f X ( x ) = otherwise. 0 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 11 / 33

  12. Uniform distribution (real case) The uniform distribution on [ 20 , 30 ] We could regard the sample space as [ 20 , 30 ] , or as all reals. 0.10 f X ( x ) � for 20 � x � 30 ; 1 / 10 f X ( x ) = 0.00 otherwise. 0 0 10 20 30 40 x � 20 � 21 . 5 21 . 5 10 dx = 0 + x 1 � P ( X � 21 . 5 ) = 0 dx + � 10 � 20 − ∞ 20 = 21 . 5 − 20 0.10 10 f X ( x ) = . 15 = 15 % 0.00 0 10 20 30 40 x Prof. Tesler Continuous Distributions Math 283 / Fall 2019 12 / 33

  13. Cumulative distribution function (CDF) The Cumulative Distribution Function (CDF) of a random variable X is F X ( x ) = P ( X � x ) For a continuous random variable, � x ′ ( x ) F X ( x ) = P ( X � x ) = − ∞ f X ( t ) dt and f X ( x ) = F X The integral cannot have “ x ” as the name of the variable in both of F X ( x ) and f X ( x ) because one is the upper limit of the integral and the other is the integration variable. So we use two variables x , t . We can either write � x F X ( x ) = P ( X � x ) = f X ( t ) dt − ∞ or � t F X ( t ) = P ( X � t ) = f X ( x ) dx − ∞ Prof. Tesler Continuous Distributions Math 283 / Fall 2019 13 / 33

  14. CDF of uniform distribution Uniform distribution on [ 20 , 30 ] � x For x < 20 : F X ( x ) = − ∞ 0 dt = 0 � 20 � x 10 dt = x − 20 1 For 20 � x < 30 : F X ( x ) = − ∞ 0 dt + 20 10 � 20 � 30 � x 1 For 30 � x : F X ( x ) = − ∞ 0 dt + 10 dt + 30 0 dt = 1 20 Together:   if x < 20 if x < 20 0 0     x − 20 1 ′ ( x )= F X ( x )= f X ( x )= F X if 20 � x � 30 if 20 � x � 30 10 10     if x � 30 if x � 30 1 0 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 14 / 33

  15. PDF vs. CDF Probability density function Cumulative distribution function 1 0.10 F X ( x ) 0.5 f X ( x ) 0 0.00 0 10 20 30 40 0 10 20 30 40 x x � F X ( x ) = . 1 if 20 � x � 30 ;  f X ( x )= if x < 20 ; 0 otherwise. 0   It’s discontinuous at x = 20 ( x − 20 ) / 10 if 20 � x � 30 ;  and 30 .  if x � 30 . 1 PDF is derivative of CDF: CDF is integral of PDF: � x ′ ( x ) f X ( x ) = F X F X ( x ) = f X ( t ) dt − ∞ Prof. Tesler Continuous Distributions Math 283 / Fall 2019 15 / 33

  16. PDF vs. CDF: Second example Probability density function Cumulative distribution function 1 density f R (r) 0.6 F R (r) 0.4 0.5 0.2 0 0 0 1 2 3 0 1 2 3 r r �  2 r / 9 if 0 � r < 3 ; if r < 0 ; 0   f R ( r )= r 2 / 9 if r � 0 or r > 3 0 F R ( r ) = if 0 � r � 3 ;  It’s discontinuous at r = 3 .  if r � 3 . 1 PDF is derivative of CDF: CDF is integral of PDF: � r ′ ( r ) f R ( r ) = F R F R ( r ) = f R ( t ) dt − ∞ Prof. Tesler Continuous Distributions Math 283 / Fall 2019 16 / 33

  17. Probability of an interval Compute P (− 1 � R � 2 ) from the PDF and also from the CDF Computation from the PDF � 2 � 0 � 2 P (− 1 � R � 2 ) = f R ( r ) dr = f R ( r ) dr + f R ( r ) dr − 1 − 1 0 � 0 � 2 2 r = 0 dr + 9 dr − 1 0 = 2 2 − 0 2 � 2 � r 2 � = 4 � = 0 + � 9 9 9 � r = 0 Computation from the CDF P (− 1 � R � 2 ) = P (− 1 − < R � 2 ) = F R ( 2 ) − F R (− 1 − ) = 2 2 9 − 0 = 4 9 Prof. Tesler Continuous Distributions Math 283 / Fall 2019 17 / 33

  18. Continuous vs. discrete random variables Cumulative distribution function Cumulative distribution function 1 1 F X (x) F R (r) 0.5 0.5 0 0 0 1 2 3 ! 1 0 1 2 r x In a continuous distribution: The probability of an individual point is 0 : P ( R = r ) = 0 . So, P ( R � r ) = P ( R < r ) , i.e., F R ( r ) = F R ( r − ) . The CDF is continuous. (In a discrete distribution, the CDF is discontinuous due to jumps at the points with nonzero probability.) P ( a < R < b )= P ( a � R < b ) = P ( a < R � b ) = P ( a � R � b ) = F R ( b ) − F R ( a ) Prof. Tesler Continuous Distributions Math 283 / Fall 2019 18 / 33

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Continuous Distributions 1.8-1.9: Continuous Random Variables 1.10.1: Uniform Distribution

Continuous Distributions 1.8-1.9: Continuous Random Variables 1.10.1: Uniform Distribution

Continuous Distributions 1.8-1.9: Continuous Random Variables 1.10.1: Uniform Distribution (Continuous) 1.10.4-5 Exponential and Gamma Distributions: Distance between crossovers Prof. Tesler Math 283 Fall 2015 Prof. Tesler Continuous

473 views • 24 slides
Chapter 5 Continuous Random Variables Continuous Probability Distributions Continuous Probability

Chapter 5 Continuous Random Variables Continuous Probability Distributions Continuous Probability

Chapter 5 Continuous Random Variables Continuous Probability Distributions Continuous Probability Distribution areas under curve correspond to probabilities for x Area A corresponds to the probability that x lies between a and b Do you see

633 views • 28 slides
Continuous Improvement Toolkit Probability Distributions Continuous Improvement Toolkit .

Continuous Improvement Toolkit Probability Distributions Continuous Improvement Toolkit .

Continuous Improvement Toolkit Probability Distributions Continuous Improvement Toolkit . www.citoolkit.com The Continuous Improvement Map Managing Selecting &amp; Decision Making Planning &amp; Project Management* Risk PDPC Break-even

614 views • 60 slides
Outline Continuous Probability Distributions The Uniform Distribution (4.1) ( ) The

Outline Continuous Probability Distributions The Uniform Distribution (4.1) ( ) The

12/14/2006 219323 Probability y and Statistics for Software and Knowledge Engineers Lecture 5: Continuous Probability Distributions Monchai Sopitkamon, Ph.D. Outline Continuous Probability Distributions The Uniform Distribution

280 views • 15 slides
Agenda Course 02402 Introduction to Statistics Continuous random variables and distributions 1

Agenda Course 02402 Introduction to Statistics Continuous random variables and distributions 1

Agenda Course 02402 Introduction to Statistics Continuous random variables and distributions 1 The Density Function The Distribution Function Lecture 3: Continuous Distributions The Mean of a Continuous Stochastic Variable The Variance of a

401 views • 8 slides
Which Distributions (or Families of Continuous . . . Distributions) Best Represent Example:

Which Distributions (or Families of Continuous . . . Distributions) Best Represent Example:

Interval Uncertainty Is . . . Data Processing . . . Interval Data . . . Family of Distributions . . . Which Distributions (or Families of Continuous . . . Distributions) Best Represent Example: Estimating . . . Interval Uncertainty: Case of

722 views • 36 slides
Continuous Probability, RVs, Distributions EECS 126 Fall 2019 September 17, 2019 Agenda

Continuous Probability, RVs, Distributions EECS 126 Fall 2019 September 17, 2019 Agenda

Continuous Probability, RVs, Distributions EECS 126 Fall 2019 September 17, 2019 Agenda Announcements Review Continuous Probability Definitions Cumulative Distribution Functions Distributions Uniform Exponential Gaussian Analogs to

761 views • 30 slides
The Gaussian Distribution Continuous distributions Probability density function (pdf) for a

The Gaussian Distribution Continuous distributions Probability density function (pdf) for a

The Gaussian Distribution Continuous distributions Probability density function (pdf) for a continuous random variable X Chris Williams, School of Informatics, University of Edinburgh b P ( a X b ) = p ( x ) dx Overview a

450 views • 5 slides
Random samples generation with Stata from continuous and discrete distributions G.

Random samples generation with Stata from continuous and discrete distributions G.

Generating random samples from Statistical Distributions Pros and cons of current functions and commands Our approach Conclusions Random samples generation with Stata from continuous and discrete distributions G. Aguilera-Venegas 1 , J.L. Gal

445 views • 13 slides
1 A simplified definition Example: Rolling a dice Let be the sample space of an experiment. A

1 A simplified definition Example: Rolling a dice Let be the sample space of an experiment. A

Outline Probabilities Conditional probabilities Bayes theorem Distributions Discrete Continuous Advanced Herd Management Distribution functions Probabilities and distributions Sampling from distributions Estimation

545 views • 10 slides
Unit 2: Probability and distributions 3. Normal and binomial distributions GOVT 3990 - Spring

Unit 2: Probability and distributions 3. Normal and binomial distributions GOVT 3990 - Spring

Unit 2: Probability and distributions 3. Normal and binomial distributions GOVT 3990 - Spring 2020 Cornell University Outline 1. Housekeeping 2. Main ideas 1. Two types of probability distributions: discrete and continuous 2. Normal

893 views • 63 slides
Testing Monotone Continuous Distributions on High-dimensional Real Cubes Michal Adamaszek DIMAP,

Testing Monotone Continuous Distributions on High-dimensional Real Cubes Michal Adamaszek DIMAP,

Introduction Monotone distributions Conclusions Testing Monotone Continuous Distributions on High-dimensional Real Cubes Michal Adamaszek DIMAP, University of Warwick Joint work with Artur Czumaj and Christian Sohler Michal Adamaszek

492 views • 38 slides
Some Continuous Distributions Normal Distribution The normal distribution with parameters and

Some Continuous Distributions Normal Distribution The normal distribution with parameters and

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control Some Continuous Distributions Normal Distribution The normal distribution with parameters and &gt; 0 has probability density function

1.24k views • 25 slides
On Learning Parametric Non-Smooth Continuous Distributions Sudeep Kamath 1 Alon Orlitsky 2

On Learning Parametric Non-Smooth Continuous Distributions Sudeep Kamath 1 Alon Orlitsky 2

On Learning Parametric Non-Smooth Continuous Distributions Sudeep Kamath 1 Alon Orlitsky 2 Venkatadheeraj Pichapati 3 Ehsan Zobeidi 2 1 PDT Partners 2 Department of Electrical and Computer Engineering University of California San Diego 3 Apple

368 views • 20 slides
Lecture 5: Probability Distributions Random Variables Probability Distributions

Lecture 5: Probability Distributions Random Variables Probability Distributions

3/26/2019 Lecture 5: Probability Distributions Random Variables Probability Distributions Discrete Random Variables Continuous Random Variables and their Distributions Discrete Joint Distributions Continuous Joint

553 views • 28 slides
ACMS 20340 Statistics for Life Sciences Chapter 11: The Normal Distributions Introducing the

ACMS 20340 Statistics for Life Sciences Chapter 11: The Normal Distributions Introducing the

ACMS 20340 Statistics for Life Sciences Chapter 11: The Normal Distributions Introducing the Normal Distributions The class of Normal distributions is the most widely used variety of continuous probability distributions. Normal density curves

382 views • 35 slides
6. Continuous Random Variables I Andrej Bogdanov Delivery time A package is to be delivered

6. Continuous Random Variables I Andrej Bogdanov Delivery time A package is to be delivered

ENGG 2430 / ESTR 2004: Probability and Statistics Spring 2019 6. Continuous Random Variables I Andrej Bogdanov Delivery time A package is to be delivered between noon and 1pm. What is the expected arrival time? Discrete model I W = { 0, 1,

949 views • 36 slides
Continuous Random Variables Recall: A continuous random variable X satisfies: its range is the

Continuous Random Variables Recall: A continuous random variable X satisfies: its range is the

ST 380 Probability and Statistics for the Physical Sciences Continuous Random Variables Recall: A continuous random variable X satisfies: its range is the union of one or more real number intervals; 1 P ( X = c ) = 0 for every c in the range of

410 views • 12 slides
Variance; Continuous Random Variables 18.05 Spring 2014 January 1, 2017 1 / 17 Variance and

Variance; Continuous Random Variables 18.05 Spring 2014 January 1, 2017 1 / 17 Variance and

Variance; Continuous Random Variables 18.05 Spring 2014 January 1, 2017 1 / 17 Variance and standard deviation X a discrete random variable with mean E ( X ) = . Meaning: spread of probability mass about the mean. Definition as

356 views • 17 slides
Alexander Kanevskiy OpenIoT Summit Europe 2016-10-12 Who am I? Alexander.Kanevskiy@Intel.com 2

Alexander Kanevskiy OpenIoT Summit Europe 2016-10-12 Who am I? Alexander.Kanevskiy@Intel.com 2

Alexander Kanevskiy OpenIoT Summit Europe 2016-10-12 Who am I? Alexander.Kanevskiy@Intel.com 2 Agenda Yocto-based distributions: platforms and products Continuous Integration and Continuous Delivery 101 Source Code Management Tools

411 views • 37 slides
CS70: Jean Walrand: Lecture 26. Continuous Probability - Pick a real number. Continuous

CS70: Jean Walrand: Lecture 26. Continuous Probability - Pick a real number. Continuous

CS70: Jean Walrand: Lecture 26. Continuous Probability - Pick a real number. Continuous Probability - Pick a random real number. Choose a real number X , uniformly at random in [ 0 , 1000 ] . What is the probability that X is exactly equal to 100

174 views • 4 slides
Multivariate random variables DS GA 1002 Statistical and Mathematical Models

Multivariate random variables DS GA 1002 Statistical and Mathematical Models

Multivariate random variables DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/DSGA1002_fall16 Carlos Fernandez-Granda Joint distributions Tool to characterize several uncertain numerical quantities of

1.39k views • 121 slides
MATH 20: PROBABILITY Sums of Independent Random Variables Xingru Chen

MATH 20: PROBABILITY Sums of Independent Random Variables Xingru Chen

MATH 20: PROBABILITY Sums of Independent Random Variables Xingru Chen xingru.chen.gr@dartmouth.edu XC 2020 Syllabus Su Sums of Random Va Variables La Law of o La Large N Numbers Ce Central Limit Theorem Ge Generating

467 views • 36 slides
Formal Modeling in Cognitive Science 1 Continuous Random Variables Lecture 21: Continuous Random

Formal Modeling in Cognitive Science 1 Continuous Random Variables Lecture 21: Continuous Random

Continuous Random Variables Continuous Random Variables Density Functions Density Functions Formal Modeling in Cognitive Science 1 Continuous Random Variables Lecture 21: Continuous Random Variables; Densities Steve Renals (notes by Frank

326 views • 4 slides

More recommend