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Continuous random Variables Anna Karlin Most Slides by Alex Tsun + - PowerPoint PPT Presentation

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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

  1. Continuous random Variables Anna Karlin Most Slides by Alex Tsun + Joshua Fan

  2. Agenda ● Recap (PDFs and CDFs) ● The (Continuous) Uniform RV ● The Exponential RV ● Memorylessness ● The normal distribution

  3. From Discrete to Continuous

  4. X cont r v PDF Intuition fxH Z

  5. PDF Intuition dw PrfXex Fxx ofxlw Fx fxH e r PHX 4 prfxex PRIX x 119111111141

  6. I to eat 0 f PRIX 4 txt PDF Intuition IT deff 3 I is l a p v k anti

  7. Itf Cumulative Distribution Functions (CDFs)

  9. 4.2 Zoo of Continuous RVs

  10. att Unifff bp I The Uniform (Continuous) RV i un if oil E E Unit Og Diego

  11. The Exponential PDF/CDF probability students Definition of Expectation n exeenbinofwtmeunB ewntsynnfd sdojt.name 1XIIJFnPoifzt 1 X until time Iwaot Exp first event

  12. P Suss event snow taxi'xIIiE The Exponential PDF/CDF a Is probability students II Definition of Prf Xlt o Prlyst Expectation at e t yH Pr PHY 1 Yet o tao Fyftko

  13. I The Exponential PDF/CDF probability students Definition of Expectation o c to Iii µ

  14. Y II fx The Exponential RV Properties dx f x y f probability students T S permin EM Van Definition of I Expectation Bga x2aE dx xhf xldx

  15. The Exponential RV Properties probability students D Definition of Expectation

  16. The Exponential RV

  17. The Exponential RV

  18. Random Picture probability students Definition of Expectation

  19. Memorylessness (Intuition) Definition of Expectation

  20. Memorylessness (Intuition) Definition of Expectation

  21. Memorylessness (Intuition) Definition of Expectation

  22. Memorylessness (Intuition) NE f o x x co o

  23. Memorylessness (Intuition) O O 1 I 25 1.5 blue orange total are we

  24. Memorylessness (Intuition)

  25. Memorylessness of Exponential (Proof) X pr X Definition of 5 Prc x Expectation pr X stt Pr X pts e

  26. Memorylessness of Exponential (Proof) Definition of Expectation

  27. i tE ENT'T example q y ● Time it takes to check someone out at a grocery store is exponential with an expected value of 10 mins. ● Independent for different customers. ● If you are the second person in line, what is the probability that you will have to wait between 10 and 20 mins. f 5 to a Tn expito t.IE IoIEho fteiEoHte 7 2 I e e qtxdx

  28. s Tamer It's example ● Time it takes to check someone out at a grocery store is exponential with an expected value of 10 mins. ● Independent for different customers ● If you are the second person in line, what is the probability that you will have to wait between 10 and 20 mins. II La La a

  29. END PIC Alex Tsun Joshua Fan

  30. A 4.3 The Normal/Gaussian Random Variable bell curve

  31. Agenda ● The Normal/Gaussian RV ● Closure properties of the Normal RV ● The standard normal CDF

  32. The Normal/Gaussian RV

  33. 10 The Normal PDF r j l I i ta XnN 4,4

  34. Random Picture probability students Definition of Expectation

  35. <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

  36. <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

  37. The Standard Normal CDF

  38. Normal Distribution Paranormal Distribution

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