1. Discrete and Continuous Distributions G. V. Narayanan

2. Discrete Probability Distributions 1.Bernoulli Probability Function 2.Binomial Probability Function 3.HyperGeometric Probability Function 4.Poisson Probability Function

3. Continuous Probability Distributions 1.Uniform Probability Function 2.Normal Probability Function 3.Standard Normal Probability Function 4.LogNormal Probability Function 5.Exponential Probability Function 6.Geometric Probability Function 7.Weibull Probability Function

4. Understand Random Variable A Random variable is a NUMERIC VALUE assigned to a ‘quantity’ or ‘property to an Object or item’ Examples of QUANTITY: Quantity can be an ‘item’ or ‘property of any object’ or ‘length’ or ‘width’ or ‘thickness’ or ‘Area’ or ‘Answers to a Questionaire’ or ‘any scientific numeric values’ Random Variable ‘Value’ can be either ‘DISCRETE’ or ‘Continuous Interval’ Mostly, a Random Variable is represented by symbol ‘X’ (Upper case Letter, never Lower case letter) Mostly, a Random Variable ‘Value’ is represented by symbol ‘x’ (Lower case letter, never Upper case letter)

5. Random Variable Values of a distribution Examples of Random Variable Values: ‘Discrete’ Random Variable Values for the toss of a COIN: Head or Tail => Assigned Values are ‘0’ (zero) for Tail (or Head) or ‘1’ (one) for Head (or Tail) ‘Discrete’ Random Variable Values for the roll of a Dice: Face up 1, Face up 2, Face up 3, Face up 4, Face up 5, Face up 6 => Assigned Values are ‘1’ (one) for Face up 1 etc Discrete Random Variable Values for the number of New cars sold is any positive number, 0,1,2,3, … Discrete Random Variable Values for the number of students is an positive number less than Max Value

6. Random Variable Values of a distribution Examples of Random Variable Values: ‘Continuous’ Random Variable Values for the height of students in a university: Any Positive Real valued number in an interval, say between (3 feet and 7 feet) with a decimal or in feet and inches ‘Continuous’ Random Variable Values for the impurities in a liquid in units of parts per millions: any Positive real values number in an interval, say between 3 and 10 PPM ‘Continuous’ Random Variable Values for the Length or Diameter of Rods: any positive real values between 0 and maximum value

7. About Population Parameters Each Probability Distribution has either ONE or TWO or Three population parameters

8. The Population Parameters of a Distribution We always talk about Either ‘Population’ Or ‘Sample’ Data from measurements or from ‘Population’ Data We will ALWAYS discuss: 1) ‘Probability’ ‘Mass’ or ‘Density’ ‘Distribution’ Functions; 2) ‘Cumulative’ Probability Distributions; 3) ‘Inverse’ Cumulative Probability Distributions For a GIVEN set of ‘Population’ Parameters These population parameters are NOT the SAME for ALL Distributions

9. The Population Parameters of a Distribution For Bernoulli Distribution, the probability of ‘success’ or ‘failure’ or ‘defective’ etc is the ONLY population parameter, denoted by symbol ‘p’ For Binomial Distribution, the TWO population parameters are: N (Total data count) and ‘p’ of Random Variable at a value For Poisson Distribution, the only population parameter is denoted by symbol ‘lamda’, this lamda equals ‘N*p’ for approximating Binomial distribution for N > 20 and p < 0.05 Note: Text treats Poisson as Discrete Distribution where as Poisson can be used for Continuous Random Variable in an interval

10. The Population Parameters of a Distribution HyperGeometric Distribution has THREE parameter: N( Total Data count), n (Sample Data count) and k (number of successes within ‘n’ selected samples of N items) The Normal Distribution has TWO population parameters: Mean value ‘mu’ and std deviation ‘sigma’. For Standard Normal Distribution, Mean = 0 and Std dev = 1

11. The Population Parameters of a Distribution LogNormal Distribution has Two Parameters: Mean and Std Dev Exponential Distribution has One Parameter Weibull Distribution has Two Parameters, Alpha and Beta

12. Discrete RV Probability Distributions 1.Binomial Distribution 2.Hypergeometric Distribution 3.Poisson Distribution 4.Geometric Distribution

13. Bernoulli Distribution Function  Bernoulli Trials are independent (assumed) p(success) = p p(Failure)=1-p  Random Variable X ~ Bernoulli(p)  Probability Mass Function of X is: p(1) = P(X=1) = p p(0) = P(X=0) = 1-p [Random Variable Values are Discrete Values ‘0’ and ‘1’]  Mean = p  Variance = p*(1-p)

14. Binomial Distribution Function

15. Geometric Distribution Function

16. Hypergeometric Distribution Function

17. Poisson Distribution Function

18. Discrete Probability Values Computing Examples

19. Computing Probability of Binomial Distribution

20. Computing Probability of Hypergeometric Distribution

21. Computing Probability of Poisson Distribution

22. Continuous RV Probability Distributions 1.Uniform Distribution 2.Standard Normal Distribution 3.Normal Distribution 4.LogNormal Distribution 5.Exponential Distribution 6.Weibull Distribution

23. Uniform Distribution Function

24. Standard Normal Distribution Function

25. Normal Distribution Function

26. LogNormal Distribution Function

27. Exponential Distribution Function

28. Weibull Distribution Function

29. Continuous Probability Values Computing Examples

30. Computing Probability of Uniform Distribution

31. Computing Probability of Standard Normal Distribution

32. Computing Probability of Normal Distribution

33. Computing Probability of LogNormal Distribution

34. Computing Probability of Exponential Distribution

35. Compute Uncertainity of Probability Distribution Mean and Variance If Population parameters are unknown, compute uncetainity on parameters computed using SAMPLE data.

36. Sample Data Values of Population Parameters If Population parameters are UNKNOWN, then Sample Data is used to compute Equivalent Population Parameters, For Example, if Mean of Population is UNKNOWN, the mean of Sample (s) can be used as equivalent to Mean of Population (Mu) Same reasoning goes for Standard Deviation value Std Dev of Sample Data can be used as Std Dev value of a population. The UNCERTANITY of error due to using Sample for obtaining Population parameters must be COMPUTED

37. Computing Uncertainty of Mean and Standard Deviation for a Binomial Distribution

38. Check on Data to find its Distribution

39. Normal Probability Plot Read Section 4.10

40. Central Limit Theorem

41. Simulation of Data for a Given Distribution

42. MiniTab Use in Computing Probability for Binomial Distribution Use Menu Calc  Probability Distributions  Binomial

43. MiniTab Use in Computing Probability for Poisson Distribution Use Menu Calc  Probability Distributions  Poisson

44. MiniTab Use in Computing Probability for Hypergeometric Distribution Use Menu Calc  ProbabilityDistributions  Hypergeomet ric See text Page 232

45. MiniTab Use in Computing Probability for Standard Normal Distribution Use Menu Calc  Probability Distributions  Normal

46. Minitab use to compute Inverse Cumulative Probability for Standard Normal Distribution Distribution Calc  Probability Distributions  Normal