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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

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