normal distribution paranormal distribution anna karlin
play

Normal Distribution Paranormal Distribution - PowerPoint PPT Presentation

Apr 06, 2024 •198 likes •699 views

Normal Distribution Paranormal Distribution Anna Karlin Most Slides by Alex Tsun Agenda The Normal/Gaussian RV Closure properties of the Normal RV The standard normal CDF The Central Limit Theorem! The

  1. Normal Distribution Paranormal Distribution Anna Karlin Most Slides by Alex Tsun

  2. Agenda ● The Normal/Gaussian RV ● Closure properties of the Normal RV ● The standard normal CDF ● The Central Limit Theorem!

  3. The Normal/Gaussian RV

  4. The Normal PDF

  5. <latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwTVu5ifEzqgxnAqelXqoxoWxMh9i1VNItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2oszYbEo2BG/5VXSrlW9i2qteVmp3+RxFOETuEcPLiCOtxBA1rAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q=</latexit> The Standard Normal CDF 0 a

  6. <latexit sha1_base64="wqZLPcGml9Og5FDdUdFUNWEF4FE=">AB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm/GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlJu2XK27VnYOsEi8nFcjR6Je/eoOYpRFKwTVu5ifEzqgxnAqelXqoxoWxMh9i1VNItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2oszYbEo2BG/5VXSrlW9i2qteVmp3+RxFOETuEcPLiCOtxBA1rAOEZXuHNeXRenHfnY9FacPKZY/gD5/MHxKuM6Q=</latexit> <latexit sha1_base64="XE2GIHEiozZRnB/MWU1YWeDNaL8=">AB8XicbVBNSwMxEJ31s9avqkcvwSLUg2W3CnosevFYwX5gu5Rsm1Ds8mSZIWy9F948aCIV/+N/+N2XYP2vpg4PHeDPzgpgzbVz321lZXVvf2CxsFbd3dvf2SweHLS0TRWiTSC5VJ8CaciZo0zDaSdWFEcBp+1gfJv57SeqNJPiwUxi6kd4KFjICDZWevTOe40Rq+CzYr9UdqvuDGiZeDkpQ45Gv/TVG0iSRFQYwrHWXc+NjZ9iZRjhdFrsJZrGmIzxkHYtFTi2k9nF0/RqVUGKJTKljBopv6eSHGk9SQKbGeEzUgvepn4n9dNTHjtp0zEiaGCzBeFCUdGoux9NGCKEsMnlmCimL0VkRFWmBgbUhaCt/jyMmnVqt5FtXZ/Wa7f5HEU4BhOoAIeXEd7qABTSAg4Ble4c3Rzovz7nzMW1ecfOYI/sD5/AF3ro95</latexit> <latexit sha1_base64="UtAQTl5HemJVA3aXQpAYqzPnUOg=">AB/nicbVDLSsNAFJ3UV62vqLhyM1iEdtGSVE3QtGNywq2FtpQbqaTdujkwcxEKHgr7hxoYhbv8Odf+MkzUJbD1w4c869zL3HjTiTyrK+jcLK6tr6RnGztLW9s7tn7h90ZBgLQtsk5KHouiApZwFtK6Y47UaCgu9y+uBOblL/4ZEKycLgXk0j6vgwCpjHCgtDcyjfmvMKjWo4its17IHVEsDs2zVrQx4mdg5KaMcrYH51R+GJPZpoAgHKXu2FSknAaEY4XRW6seSRkAmMKI9TQPwqXSbP0ZPtXKEHuh0BUonKm/JxLwpZz6ru70QY3lopeK/3m9WHmXTsKCKFY0IPOPvJhjFeI0CzxkghLFp5oAEUzviskYBClE0tDsBdPXiadRt0+qzfuzsvN6zyOIjpGJ6iCbHSBmugWtVAbEZSgZ/SK3own48V4Nz7mrQUjnzlEf2B8/gAFuJLx</latexit> <latexit sha1_base64="UyhIi9uJIiIGu7TMPZg2VlqWys=">AB8HicbVBNSwMxEJ31s9avqkcvwSLUg2W3CnosevFYwX5Iu5Rsm1Dk+ySZIWy9Fd48aCIV3+ON/+N2XYP2vpg4PHeDPzgpgzbVz321lZXVvf2CxsFbd3dvf2SweHLR0litAmiXikOgHWlDNJm4YZTjuxolgEnLaD8W3mt5+o0iySD2YSU1/goWQhI9hY6bHXGLHKOT4r9ktlt+rOgJaJl5My5Gj0S1+9QUQSQaUhHGvd9dzY+ClWhFOp8VeomMyRgPadSiQXVfjo7eIpOrTJAYaRsSYNm6u+JFAutJyKwnQKbkV70MvE/r5uY8NpPmYwTQyWZLwoTjkyEsu/RgClKDJ9Ygoli9lZERlhYmxGWQje4svLpFWrehfV2v1luX6Tx1GAYziBCnhwBXW4gwY0gYCAZ3iFN0c5L8678zFvXHymSP4A+fzBweBjz4=</latexit> The Standard Normal CDF Φ ( − a ) = 1 − Φ ( a ) Φ ( − a ) 1 − Φ ( a ) 0 a

  7. The Standard Normal CDF

  8. What about non-Standard Normals?

  9. We can Standardize any RV probability students Definition of Expectation

  10. Normals stay normal! (Under scale+Shift) probability students Definition of Expectation

  11. Closure of the normal (Under scale+Shift) probability students Definition of Expectation

  12. X is normal with mean 3 and variance 9. What is Pr (2 < X < 5) ○ Pr (X > 0) ○ Pr (|X-3| > 6) ○

  13. X is normal with mean 3 and variance 9. What is Pr (2 < X < 5) ○ Pr (X > 0) ○ Pr (|X-3| > 6) ○

  14. <latexit sha1_base64="N0Sk5aodJRL9coO8QfPWdqzEbFg=">AB+nicbVDLSsNAFJ34rPWV6tLNYBEqSEmqoMuiG1dSwT6giWUynbRDZyZhZqKU2E9x40IRt36JO/GSZuFth64cDjnXu69J4gZVdpxvq2l5ZXVtfXCRnFza3tn1y7tVSUSEyaOGKR7ARIEUYFaWqGenEkiAeMNIORleZ34gUtFI3OlxTHyOBoKGFCNtpJ5duql4PDmBnqIDju5rx8WeXaqzhRwkbg5KYMcjZ795fUjnHAiNGZIqa7rxNpPkdQUMzIpeokiMcIjNCBdQwXiRPnp9PQJPDJKH4aRNCU0nKq/J1LElRrzwHRypIdq3svE/7xuosMLP6UiTjQReLYoTBjUEcxygH0qCdZsbAjCkpbIR4ibA2aWUhuPMvL5JWreqeVmu3Z+X6ZR5HARyAQ1ABLjgHdXANGqAJMHgEz+AVvFlP1ov1bn3MWpesfGYf/IH1+QMCAZKJ</latexit> From to standard normal N ( µ, σ 2 )

  15. Summary: The Normal/Gaussian RV

  16. Normal random variables

  17. Closure of the normal (under addition) probability students Definition of Expectation

  18. Closure of the normal (under addition) probability students Definition of Expectation

  19. 5.7 The Central Limit Theorem

  20. The Sample Mean

  21. The Sample Mean

  22. The Central Limit Theorem Consider i.i.d. (independent, identically distributed) random vars X 1 , X 2 , X 3 , … Where X i has μ = E[X i ] and σ 2 = Var[X i ] Consider random variables X 1 + X 2 + . . . + X n and n 1 X X i n i =1

  23. The Central Limit Theorem Consider i.i.d. (independent, identically distributed) random vars X 1 , X 2 , X 3 , … Where X i has μ = E[X i ] and σ 2 = Var[X i ] As n → ∞, n µ, σ 2 ✓ ◆ M n = 1 X Restated: As n → ∞, X i → N n n i =1

  24. CLT (Pictures) Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

  25. CLT (Pictures) Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

  26. CLT (Pictures) Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

  27. CLT (Pictures) Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

  28. CLT (Pictures) Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

  29. CLT (Pictures) Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

  30. CLT (Pictures) Fr From: https://courses.cs.washington.edu/courses/cse312/17wi/slides/10limits.pdf

  31. CLT in the real world CLT is the reason many things appear normally distributed Many quantities = sums of (roughly) independent random vars Exam scores: sums of individual problems People’s heights: sum of many genetic & environmental factors Measurements: sums of various small instrument errors

  32. CLT in the real world

  33. CLT in the real world

  34. CLT in the real world

  35. CLT in the real world

  36. CLT in the real world

  37. CLT (Example) Definition of Expectation

  38. CLT (Example) Definition of Expectation

  39. CLT (Example) Definition of Expectation

  40. CLT (Example) Definition of Expectation

  41. The Continuity Correction (Idea) ● Suppose I asked you to estimate Pr (X = 20) using the normal approximation. ● Problem: Binomial is discrete, Normal is continuous.

  42. The Continuity Correction (Idea) Definition of Expectation

  43. The Continuity Correction (Idea) Definition of Expectation

  44. The Continuity Correction (Idea) Definition of Expectation

  45. The Continuity Correction

  46. The Continuity Correction

  47. The Central Limit Theorem

  48. Normal random variables

  49. The Standard Normal CDF

Recommend

1. Normal distribution 2. Geometric distribution 3. Binomial distribution 4.

1. Normal distribution 2. Geometric distribution 3. Binomial distribution 4.

Chapter 3. Distribution of Random Variables Jan 26, 2016 by Huamei Dong 1. Normal distribution 2. Geometric distribution 3. Binomial distribution 4. Negative binomial distribution 5. Poisson distribution 1.Normal distribution

520 views • 11 slides
Continuous random Variables Anna Karlin Most Slides by Alex Tsun + Joshua Fan Agenda Recap

Continuous random Variables Anna Karlin Most Slides by Alex Tsun + Joshua Fan Agenda Recap

Continuous random Variables Anna Karlin Most Slides by Alex Tsun + Joshua Fan Agenda Recap (PDFs and CDFs) The (Continuous) Uniform RV The Exponential RV Memorylessness The normal distribution From Discrete to Continuous X

517 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
The Normal Distribution INFO-1301, Quantitative Reasoning 1 University of Colorado Boulder March

The Normal Distribution INFO-1301, Quantitative Reasoning 1 University of Colorado Boulder March

The Normal Distribution INFO-1301, Quantitative Reasoning 1 University of Colorado Boulder March 17, 2017 Prof. Michael Paul Normal Distribution Normal Distribution Normal Distribution The most common curve in all of statistics and in

285 views • 15 slides
U nit 2: P robability and distributions L ecture 2: N ormal distribution S tatistics 101 Mine C

U nit 2: P robability and distributions L ecture 2: N ormal distribution S tatistics 101 Mine C

U nit 2: P robability and distributions L ecture 2: N ormal distribution S tatistics 101 Mine C etinkaya-Rundel September 17, 2013 Normal distribution Normal distribution 1 Normal distribution model 68-95-99.7 Rule Standardizing with Z

1.18k views • 71 slides
1.10.2 Normal distribution 1.10.3 Approximating binomial distribution by normal 2.10 Central

1.10.2 Normal distribution 1.10.3 Approximating binomial distribution by normal 2.10 Central

1.10.2 Normal distribution 1.10.3 Approximating binomial distribution by normal 2.10 Central Limit Theorem Prof. Tesler Math 283 Fall 2019 Prof. Tesler 1.10.2-3, 2.10 Normal distribution Math 283 / Fall 2019 1 / 38 Normal distribution

755 views • 38 slides
The Normal Distribution The normal distribution plays a central role in probability theory and in

The Normal Distribution The normal distribution plays a central role in probability theory and in

ST 380 Probability and Statistics for the Physical Sciences The Normal Distribution The normal distribution plays a central role in probability theory and in statistics. It is often used as a model for the distribution of continuous random

197 views • 16 slides
The Normal Distribution Part 2: Standardization and Percentiles INFO-1301, Quantitative Reasoning

The Normal Distribution Part 2: Standardization and Percentiles INFO-1301, Quantitative Reasoning

The Normal Distribution Part 2: Standardization and Percentiles INFO-1301, Quantitative Reasoning 1 University of Colorado Boulder March 20, 2017 Prof. Michael Paul Normal Distribution What can we do with this? If the normal distribution is

402 views • 17 slides
Slides Added over Course 2 Normal Distribution Tutorial (1) Let Q be the order quantity, and

Slides Added over Course 2 Normal Distribution Tutorial (1) Let Q be the order quantity, and

Slides Added over Course 2 Normal Distribution Tutorial (1) Let Q be the order quantity, and Start with 0.0180 Center the ( , ) the parameters of the = 100, 0.0160 distribution over 0 normal demand distribution 0.0140 =

361 views • 9 slides
Exponential &amp; Normal Distribution Lec.22 July 29, 2020 Exponential Distribution: Fundamental

Exponential &amp; Normal Distribution Lec.22 July 29, 2020 Exponential Distribution: Fundamental

Exponential &amp; Normal Distribution Lec.22 July 29, 2020 Exponential Distribution: Fundamental Idea The exponential distribution is the continuous analog of the geometric distribution. In the case of the geometric coin flipping experiment,

372 views • 33 slides
JUST THE MATHS SLIDES NUMBER 19.8 PROBABILITY 8 (The normal distribution) by A.J.Hobson

JUST THE MATHS SLIDES NUMBER 19.8 PROBABILITY 8 (The normal distribution) by A.J.Hobson

JUST THE MATHS SLIDES NUMBER 19.8 PROBABILITY 8 (The normal distribution) by A.J.Hobson 19.8.1 Limiting position of a frequency polygon 19.8.2 Area under the normal curve 19.8.3 Normal distribution for continuous variables UNIT 19.8

402 views • 14 slides
4.3 Normal distribution Prof. Tesler Math 186 Winter 2020 Prof. Tesler 4.3 Normal distribution

4.3 Normal distribution Prof. Tesler Math 186 Winter 2020 Prof. Tesler 4.3 Normal distribution

4.3 Normal distribution Prof. Tesler Math 186 Winter 2020 Prof. Tesler 4.3 Normal distribution Math 186 / Winter 2020 1 / 40 Normal distribution a.k.a. Bell curve and Gaussian distribution The normal distribution is a continuous

670 views • 40 slides
Unit 2: Probability and distributions Lecture 3: Normal distribution Statistics 101 Thomas

Unit 2: Probability and distributions Lecture 3: Normal distribution Statistics 101 Thomas

Unit 2: Probability and distributions Lecture 3: Normal distribution Statistics 101 Thomas Leininger May 23, 2013 Announcements Announcements 1 Normal distribution 2 Normal distribution model 68-95-99.7 Rule Standardizing with Z scores

706 views • 67 slides
Lecture 5: The multivariate normal distribution The bivariate normal distribution Suppose x ,

Lecture 5: The multivariate normal distribution The bivariate normal distribution Suppose x ,

Lecture 5: The multivariate normal distribution The bivariate normal distribution Suppose x , y , x 0, y 0 and 1 1 are constants. Define the 2 2 matrix by 2 x y x = . 2 x

934 views • 24 slides
Outline Probability Calculations Using the Normal Distribution (5.1) Linear Combinations

Outline Probability Calculations Using the Normal Distribution (5.1) Linear Combinations

12/21/2006 219323 Probability y and Statistics for Software and Knowledge Engineers Lecture 6: The Normal Distribution Monchai Sopitkamon, Ph.D. Outline Probability Calculations Using the Normal Distribution (5.1) Linear

590 views • 20 slides
MATH 105: Finite Mathematics 9-6: The Normal Distribution Prof. Jonathan Duncan Walla Walla

MATH 105: Finite Mathematics 9-6: The Normal Distribution Prof. Jonathan Duncan Walla Walla

Bernoulli Probability and the Normal Distribution Properties of the Normal Distribution Examples Conclusion MATH 105: Finite Mathematics 9-6: The Normal Distribution Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Bernoulli

890 views • 63 slides
Introductory Statistics Day 15 Introduction to the Normal Distribution Big ideas about normal

Introductory Statistics Day 15 Introduction to the Normal Distribution Big ideas about normal

Introductory Statistics Day 15 Introduction to the Normal Distribution Big ideas about normal distributions In a probability distribution, the total area represented in the graph is always 1 (or 100%) and the probability of being in a given

729 views • 23 slides
Lecture 3: The Normal Distribution and Statistical Inference Ani Manichaikul amanicha@jhsph.edu

Lecture 3: The Normal Distribution and Statistical Inference Ani Manichaikul amanicha@jhsph.edu

Lecture 3: The Normal Distribution and Statistical Inference Ani Manichaikul amanicha@jhsph.edu 19 April 2007 1 / 62 A Review and Some Connections The Normal Distribution The Central Limit Theorem Estimates of means and proportions: uses

907 views • 62 slides
Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda

Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda

Anonymous questions Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda Intro to Discrete Random Variables Probability Mass Functions Cumulative Distribution function Expectation

643 views • 24 slides
Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda

Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda

Probability 3.1 Discrete Random Variables Basics Anna Karlin Most slides by Alex Tsun Agenda Intro to Discrete Random Variables Probability Mass Functions Cumulative Distribution function Expectation Flipping two coins Random

738 views • 36 slides
The Normal Distribution: The Normal curve is a mathematical abstraction which conveniently

The Normal Distribution: The Normal curve is a mathematical abstraction which conveniently

The Normal Distribution: The Normal curve is a mathematical abstraction which conveniently describes (&quot;models&quot;) many frequency distributions of scores in real-life. length of time before someone looks away in a staring contest:

579 views • 29 slides
Mathematical Foundations for Finance Exercise 8 Martin Stefanik ETH Zurich Normal Distribution

Mathematical Foundations for Finance Exercise 8 Martin Stefanik ETH Zurich Normal Distribution

Mathematical Foundations for Finance Exercise 8 Martin Stefanik ETH Zurich Normal Distribution 2 1 Due to the definition of Brownian motion, normal distribution plays an of X respectively , we have that cannot be expressed in a closed form.

479 views • 8 slides
Distribution The definition of distribution Distribution of the subject-term Distribution of the

Distribution The definition of distribution Distribution of the subject-term Distribution of the

Three Acts of the Mind Verbal Expression: Mental Act: Simple Apprehension Term Judgment Proposition Deductive Inference Syllogism Slide 8-1 Distribution The definition of distribution Distribution of the subject-term

390 views • 12 slides
Properties of the Normal Distribution MDM4U: Mathematics of Data Management Recap For a normal

Properties of the Normal Distribution MDM4U: Mathematics of Data Management Recap For a normal

n o r m a l d i s t r i b u t i o n n o r m a l d i s t r i b u t i o n Properties of the Normal Distribution MDM4U: Mathematics of Data Management Recap For a normal distribution with = 17 and = 2, what is the likelihood that a datum has

379 views • 3 slides

More recommend