1. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §11.3 Variance, Expected-Value

2. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §11.2 Continuous Probability  Any QUESTIONS About HomeWork §11.2 → HW-21 11.2

3. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 3 Bruce Mayer, PE Chabot College Mathematics §11.3 Learning Goals  Compute and use expected value  Interpret variance and standard deviation  Find expected value for a joint probability density function

4. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 4 Bruce Mayer, PE Chabot College Mathematics Continuous PDF Expected Value  ReCall DISCRETE Random Variable, X, Probability P Distribution  The EXPECTED VALUE, or average, by a Weighted Average Calculation X → x1x1 x2x2 x4x4 … xnxn P → p1p1 p2p2 p4p4 … pnpn

5. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 5 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 6 Bruce Mayer, PE Chabot College Mathematics Continuous PDF Expected Value  If the “Averaging” Interval, or Domain, expands to ±∞  Quick Example: consider the Probability Distribution Function:

7. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 7 Bruce Mayer, PE Chabot College Mathematics Continuous PDF Expected Value  SubStitute the PDF into the Expected Value Integral  Thus the Average of the Random Variable is 2/3

8. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 9 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  CellPh Battery Life  Substituting in u & dv  Thus the CellPh will have average tolerable lifespan is about 0.893 years (~326 days).

11. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 11 Bruce Mayer, PE Chabot College Mathematics Continuous PDF: Var & StdDev  ReCall the Variance of a DISCRETE Random Variable  Again using:  Find  And Standard Deviation:

12. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 13 Bruce Mayer, PE Chabot College Mathematics Simplify Variance Formula  ReCall a Property of ANY PDF:  Also by DEFINITION for a PDF  Using the Above in the Var Equation

14. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 14 Bruce Mayer, PE Chabot College Mathematics Simplify Variance Formula  Simplify the Last Equation  Thus the Simplified Formula

15. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Expected Value  Consider the Probability Distribution Function  Then the Expected Value for X

16. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Expected Value  And the Variance  Then the Standard Deviation

17. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 17 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 18 Bruce Mayer, PE Chabot College Mathematics 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 ): 2.

19. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 19 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Joint PDF  Consider this Joint (BiVariate) Probability Distribution function 1.Verify that this is a Valid PDF 2.Calculate P(X ≤ 2,Y ≤ ½ ) 3.Compute E(X) = µ X 4.Compute E(Y) = µ Y

21. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Joint PDF  Verify PDF a.For the Given POSITIVE Domains the function 6xy 2 is AlWays NONnegative  b.Check Integration to One –as integration is ZERO outside of the 0≤ x,y ≤1 domain

22. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Joint PDF  Complete Computations  Thus in this case 

23. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Joint PDF  Calculate P(X ≤ 2,Y ≤ ½ )  Thus P(X ≤ 2,Y ≤ ½ ) = 1/8 = 12.5%

24. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Joint PDF  Find E ( X ) = µ X  Thus E ( X ) = µ X = 2/3

25. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Joint PDF  Find E ( Y ) = µ Y  Thus E ( Y ) = µ Y = ¾

26. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 26 Bruce Mayer, PE Chabot College Mathematics WhiteBoard PPT Work  Problems From §11.3 P26 → Rat Maze

27. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 27 Bruce Mayer, PE Chabot College Mathematics All Done for Today Exponential PDF

28. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 28 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

29. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 29 Bruce Mayer, PE Chabot College Mathematics

30. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 30 Bruce Mayer, PE Chabot College Mathematics

31. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 31 Bruce Mayer, PE Chabot College Mathematics

32. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 32 Bruce Mayer, PE Chabot College Mathematics

33. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 33 Bruce Mayer, PE Chabot College Mathematics

34. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 34 Bruce Mayer, PE Chabot College Mathematics

35. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 35 Bruce Mayer, PE Chabot College Mathematics

36. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 36 Bruce Mayer, PE Chabot College Mathematics 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)