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 §11.2 Probability Distribution Fcns

2. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §11.1 Discrete Probability  Any QUESTIONS About HomeWork §11.1 → HW-20 11.1

3. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 3 Bruce Mayer, PE Chabot College Mathematics §11.2 Learning Goals  Define and examine continuous probability density/distribution functions  Use uniform and exponential probability distributions  Study joint probability distributions

4. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 4 Bruce Mayer, PE Chabot College Mathematics Probability Distribution  Consider Data on the Height of a sample group of 20 year old Men  We can Plot this Frequency Data using bar y_abs=[1,0,0,0,2,4,5,4,8,11,12,1 0,9,8,7,5,4,4,3,1,1,0,1] xbins = [64:0.5:75]; axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green bar(xbins, y_abs, 'LineWidth', 2),grid,... xlabel('\fontsize{14}Height (Inches)'), ylabel('\fontsize{14}Height (Inches)'),... title(['\fontsize{16}Height of 20 Yr-Old Men',])

5. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 5 Bruce Mayer, PE Chabot College Mathematics Probability Distribution Fcn (PDF)  Because the Area Under the Scaled Plot is 1.00, exactly, The FRACTIONAL Area under any bar, or set-of-bars gives the probability that any randomly Selected 20 yr-old man will be that height  e.g., from the Plot we Find 67.5 in → 4% 68 in → 8% 68.5 in → 11%  Summing → 23 %  Thus by this data- set 23% of 20 yr-old men are 67.25- 68.75 inches tall

6. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 6 Bruce Mayer, PE Chabot College Mathematics Random variables can be Discrete or Continuous  Discrete random variables have a countable number of outcomes Examples: Dead/Alive, Red/Black, Heads/Tales, dice, deck of cards, etc.  Continuous random variables have an infinite continuum of possible values. Examples: Battery Current, human weight, Air Temperature, the speed of a car, the real numbers from 7 to 11.

7. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 7 Bruce Mayer, PE Chabot College Mathematics Continuous Case  The probability function that accompanies a continuous random variable is a continuous mathematical function that integrates to 1.  The Probabilities associated with continuous functions are just areas under a Region of the curve (→ Definite Integrals)  Probabilities are given for a range of values, rather than a particular value e.g., the probability of Jan RainFall in Hayward, CA being between 6-7 inches (avg = 5.20”)

8. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 8 Bruce Mayer, PE Chabot College Mathematics Continuous Probability Dist Fcn

9. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 9 Bruce Mayer, PE Chabot College Mathematics Continuous Case PDF Example  Recall the negative exponential function (in probability, this is called an “exponential distribution”):  This Function Integrates to 1 for limits of zero to infinity as required for all PDF’s

10. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 10 Bruce Mayer, PE Chabot College Mathematics Continuous Case PDF Example x p(x)=e -x 1  For example, the probability of x falling within 1 to 2:  The probability that x is any exact value (e.g.: 1.9476) is 0 we can ONLY assign Probabilities to possible RANGES of x x 1 12 p(x)=e -x NO Area Under a LINE

11. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  DownLoad Wait  When downloading OpenProject SoftWare, the website may put users in a queue as they attempt the download.  The time spent in line before the particular download begins is a random variable with approx. density function

12. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  DownLoad Wait  For this PDF then, What is the probability that a user waits at least five (5) minutes before the download?  SOLUTION:  We need P(x) ≥ 5 which can be found by integration and noting that if x is larger than 10, the probability is zero. Thus by the Probability:

13. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  DownLoad Wait  Continue PDF Reduction  Thus There is a 43.75% chance of a 5 minute PreDownLoad Wait Time

14. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Build a PDF  Find a value of k so that the following represents a Valid, Continuous Probability Distribution Function

15. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Build a PDF  SOLUTION:  The function is always NON-negative for non-negative inputs, so simply need to verify that the definite integral equals 1 (that all probabilities together Add-Up, or Integrate, to 100%).  Thus, the correct value of k produces this functional behavior →

16. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Build a PDF  Because the function is identically zero everywhere outside of the interval [ 0, k ], restrict the evaluation to that interval →  Solve by SubStitution; Let:  Then

17. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Build a PDF  Then

18. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Build a PDF  Finally  However, the 0 ≤ x ≤ k interval ends in a non-negative value so need k -positive:  Thus the Desired PDF

19. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 19 Bruce Mayer, PE Chabot College Mathematics Uniform Density Function  Definition  Graph

20. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Random No. Generator  A Random Number Generator (RNG) selects any number between 0 and 100 (including any number of decimal places).  Because each number is equally likely, a uniform distribution models the probability distribution.  What is the probability that the RNG selects a number between 50 and 60?

21. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Random No. Generator  SOLUTION: The Probability Distribution Function:  Then the Probability of Generating a RN between 50 & 60

22. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Random No. Generator  Evaluating the Integral  As Expected find the Probability of a 50-60 RN as 10%

23. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 23 Bruce Mayer, PE Chabot College Mathematics Exponential Density Function  Definition  Graph

24. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  SmartPhone LifeSpan  The battery of a popular SmartPhone loses about 20% of its charged capacity after 400 full charges.  Assuming one charge per day, the estimated probability density function for the length of tolerable lifespan for a phone that is t years old →

25. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  SmartPhone LifeSpan  Find the probability that the tolerable lifespan of the SmartPhone is at least 500 days (500 charges).  SOLUTION: The probability of a tolerable lifespan being greater than or equal to 500 days (500/365 years):

26. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 26 Bruce Mayer, PE Chabot College Mathematics Joint Probability Distribution Fcn  A joint probability density function f(x, y) has the following properties: 1. f(x, y) ≥ 0 for all points (x, y) in the Cartesian Plane 2.Double Integrates to 1: 3.The Probability that an Ordered Pair, (X, Y) Lies in Region R found by:

27. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 27 Bruce Mayer, PE Chabot College Mathematics Joint Probability Distribution Fcn  Example joint probability density function Graph

28. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  Joint PDF  Consider the Joint PDF:  Find:

29. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 29 Bruce Mayer, PE Chabot College Mathematics WhiteBoard PPT Work  Problems From §11.2 P48 → Traffic Lite Roullette

30. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 30 Bruce Mayer, PE Chabot College Mathematics All Done for Today Fitting PDFs to Hists

31. BMayer@ChabotCollege.edu MTH16_Lec-21_sec_11-2_Continuous_PDFs.pptx 31 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

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