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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

Chabot Mathematics. §11.3 Variance, Expected-Value. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 11.2. Review §. Any QUESTIONS About §11.2 Continuous Probability Any QUESTIONS About HomeWork §11.2 → HW-21. §11.3 Learning Goals.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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  1. Chabot Mathematics §11.3 Variance,Expected-Value Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 11.2 Review § • Any QUESTIONS About • §11.2 Continuous Probability • Any QUESTIONS About HomeWork • §11.2 → HW-21

  3. §11.3 Learning Goals • Compute and use expected value • Interpret variance and standard deviation • Find expected value for a joint probability density function

  4. Continuous PDF Expected Value • ReCall DISCRETE Random Variable, X, Probability P Distribution • The EXPECTED VALUE, or average, by a Weighted Average Calculation

  5. Continuous PDF Expected Value • Compare to the Discrete Case a CONTINUOUS PDF described by the Function f(x) • Then the Expected Value of the PDF

  6. Continuous PDF Expected Value • If the “Averaging” Interval, or Domain, expands to ±∞ • Quick Example: consider the Probability Distribution Function:

  7. Continuous PDF Expected Value • SubStitute the PDF into the Expected Value Integral • Thus the Average of the Random Variable is 2/3

  8. Example  CellPh Battery Life • The battery of a popular smartphone loses about 20% of its charged capacity after 400 full charges. Assuming one charge per day, then the estimated PDF for the length of tolerable lifespan for a phone that is t years old → • For this Phone Find the Average Tolerable LifeSpan

  9. Example  CellPh Battery Life • SOLUTION: • The Average Tolerable LifeSpan is given by the expected value of the random variable T: • Integrate by PARTS Using

  10. Example  CellPh Battery Life • Substituting in u & dv • Thus the CellPh will have average tolerable lifespan is about 0.893 years (~326 days).

  11. Continuous PDF: Var & StdDev • ReCall the Variance of a DISCRETE Random Variable • Again using: • Find • And Standard Deviation:

  12. Simplify Variance Formula • First Let E(X) = µ • Then • Now Expand (Multiply-out) [x−µ]2 • Distribute f(x), and Integrate Term-by-Term, noting that µ is a CONSTANT

  13. Simplify Variance Formula • ReCall a Property of ANY PDF: • Also by DEFINITION for a PDF • Using the Above in the Var Equation

  14. Simplify Variance Formula • Simplify the Last Equation • Thus the Simplified Formula

  15. Example  Expected Value • Consider the Probability Distribution Function • Then the Expected Value for X

  16. Example  Expected Value • And the Variance • Then the Standard Deviation

  17. Joint PDF • Consider X & Y continuous random variables whose Probability Distribution Function depends simultaneously on values of Both x, y; that is • For any valid Region A defined by a combination of X & Y

  18. Joint PDF Properties • Joint PDF exhibit the same behavior as single-variable PDF’s • In summary, for any valid input Region, R, for the Joint PDF f(x,y):

  19. Joint PDF Expected-Value • Calculate E(X) and E(Y) for the Joint PDF using the same Weighted-Average Method as used for Single Variable PDF • And the Y version

  20. Example  Joint PDF • Consider this Joint (BiVariate) Probability Distribution function • Verify that this is a Valid PDF • Calculate P(X ≤ 2 ,Y≤ ½ ) • Compute E(X) = µX • Compute E(Y) = µY

  21. Example  Joint PDF • Verify PDF • For the Given POSITIVE Domains the function 6xy2 is AlWaysNONnegative • Check Integration to One • as integration is ZERO outside of the 0≤x,y≤1 domain

  22. Example  Joint PDF • Complete Computations • Thus in this case 

  23. Example  Joint PDF • Calculate P(X ≤ 2 ,Y ≤ ½ ) • Thus P(X ≤ 2 ,Y ≤ ½ ) = 1/8 = 12.5%

  24. Example  Joint PDF • Find E(X) = µX • Thus E(X) = µX = 2/3

  25. Example  Joint PDF • Find E(Y) = µY • Thus E(Y) = µY = ¾

  26. WhiteBoard PPT Work • Problems From §11.3 • P26 → Rat Maze

  27. All Done for Today ExponentialPDF

  28. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

  29. By MuPAD • Uf := int(a*x*E^(-b*x), x) • assume(b, Type::Positive): • U1 := int(a*x*E^(-b*x), x=0..infinity) • U2 := int(a*x*x*E^(-b*x), x=0..infinity) • P := int((1/9)*x*E^(-(1/3)*x), x=5..7) • Pnum = float(P)

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