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

1. Chabot Mathematics §11.2 ProbabilityDistribution Fcns Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 11.1 Review § • Any QUESTIONS About • §11.1 Discrete Probability • Any QUESTIONS About HomeWork • §11.1 → HW-20

3. §11.2 Learning Goals • Define and examine continuous probability density/distribution functions • Use uniform and exponential probability distributions • Study joint probability distributions

4. Consider Data on the Height of a sample group of 20 year old Men Probability Distribution • We can Plot this Frequency Data using bar y_abs=[1,0,0,0,2,4,5,4,8,11,12,10,9,8,7,5,4,4,3,1,1,0,1] 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. 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 Probability Distribution Fcn (PDF)

6. 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. 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. Continuous Probability Dist Fcn

9. 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. 1 2 Continuous Case PDF Example • The probability that x is any exact value (e.g.: 1.9476) is 0 • we can ONLY assign Probabilities to possible RANGES of x • For example, the probability of x falling within 1 to 2: p(x)=e-x 1 x p(x)=e-x NO Area Under a LINE 1 x

11. Example  DownLoad Wait • When downloading OpenProjectSoftWare, 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. 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. Example  DownLoad Wait • ContinuePDFReduction • Thus There is a 43.75% chance of a 5 minute PreDownLoad Wait Time

14. Example  Build a PDF • Find a value of k so that the following represents a Valid, Continuous Probability Distribution Function

15. 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. 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. Example  Build a PDF • Then

18. 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. Uniform Density Function • Definition • Graph

20. 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. Example  Random No. Generator • SOLUTION: The Probability Distribution Function: • Then the Probability of Generating a RN between 50 & 60

22. Example  Random No. Generator • Evaluating the Integral • As Expected find the Probability of a 50-60 RN as 10%

23. Exponential Density Function • Definition • Graph

24. Example  SmartPhoneLifeSpan • 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. Example  SmartPhoneLifeSpan • Find the probability that the tolerable lifespan of the SmartPhoneis 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. Joint Probability Distribution Fcn • A joint probability density functionf(x, y) has the following properties: • f(x, y) ≥ 0 for all points (x, y) in the Cartesian Plane • Double Integrates to 1: • The Probability that an Ordered Pair, (X, Y)Lies in Region R found by:

27. Joint Probability Distribution Fcn • Example joint probability density function Graph

28. Example  Joint PDF • Consider the Joint PDF: • Find:

29. WhiteBoard PPT Work • Problems From §11.2 • P48 → Traffic LiteRoullette

30. All Done for Today FittingPDFs toHists

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