A. A. Elimam College of Business San Francisco State University

1. Random Variables And Probability Distributions A. A. Elimam College of Business San Francisco State University

2. Topics • Basic Probability Concepts: • Sample Spaces and Events, Simple • Probability, and Joint Probability, • Conditional Probability • Random Variables • Probability Distributions • Expected Value and Variance of a RV

3. Topics • Discrete Probability Distributions • Bernoulli and Binomial Distributions • Poisson Distributions • Continuous Probability Distributions • Uniform • Normal • Triangular • Exponential

4. Topics • Random Sampling and Probability Distributions • Random Numbers • Sampling from Probability Distributions • Simulation Techniques • Sampling Distributions and Sampling Errors • Use of Excel

5. Probability 1 Certain • Probability is the likelihood • that the event will occur. • Two Conditions: • Value is between 0 and 1. • Sum of the probabilities of all events must be 1. .5 0 Impossible

6. Probability: Three Ways • First: Process Generating Events is Known: Compute using classical probability definition • Example: rolling dice • Second: Relative Frequency: Compute using empirical data • Example: Rain Next day based on history • Third: Subjective Method: Compute based on judgment • Example: Analyst predicts DIJA will increase by 10% over next year

7. Random Variable • A numerical description of the outcome of an experiment • Example:Discrete RV: countable # of outcomes • Throw a die twice: Count the number of times 4 comes up (0, 1, or 2 times)

8. Discrete Random Variable • Discrete Random Variable: • Obtained by Counting (0, 1, 2, 3, etc.) • Usually finite by number of different values • e.g. • Toss a coin 5 times. Count the number of tails.(0, 1, 2, 3, 4, or 5 times)

9. Random Variable • A numerical description of the outcome of an experiment • Example: Continuous RV: • The Value of the DJIA • Time to repair a failed machine • RV Given by Capital Letters X & Y • Specific Values Given by lower case

10. Probability Distribution • Characterization of the possible values that a RV may assume along with the probability of assuming these values.

11. Discrete Probability Distribution • List of all possible [ xi, p(xi) ] pairs • Xi = value of random variable • P(xi) = probability associated with value • Mutually exclusive (nothing in common) • Collectively exhaustive (nothing left out) • 0 p(xi)  1 • P(xi) = 1

12. Weekly Demand of a Slow-Moving Product Probability Mass Function

13. Weekly Demand of a Slow-Moving Product A Cumulative Distribution Function: Probability that RV assume a value <= a given value, x

14. Sample Spaces Collection of all Possible Outcomes e.g. All 6 faces of a die: e.g. All 52 cards of a bridge deck:

15. Events • Simple Event:Outcome from a Sample Space • with 1 Characteristic • e.g. A Red Card from a deck of cards. • Joint Event: Involves 2 Outcomes Simultaneously • e.g. An Acewhich is also aRed Cardfrom a • deck of cards.

16. Visualizing Events • Contingency Tables • Tree Diagrams Ace Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52

17. Simple Events The Event of a Happy Face There are 5 happy faces in this collection of 18 objects

18. Joint Events The Event of a Happy Face AND Light Colored 3Happy Faces which are light in color

19.   Special Events Null Event • Null event • Club & diamond on 1 card draw • Complement of event • For event A, • All events not In A:

20. Dependent or Independent Events The Event of a Happy Face GIVEN it is Light Colored E = Happy FaceLight Color 3Items:3 Happy Faces Given they are Light Colored

21. Contingency Table A Deck of 52 Cards Red Ace Not an Ace Total Ace Red 2 24 26 Black 2 24 26 Total 4 48 52 Sample Space

22. Tree Diagram Event Possibilities Ace Red Cards Not an Ace Full Deck of Cards Ace Black Cards Not an Ace

23. Computing Probability • The Probability of an Event, E: • Each of the Outcome in the Sample Space • equally likely to occur. Number of Event Outcomes P(E) = Total Number of Possible Outcomes in the Sample Space X e.g. P() = 2/36 = T (There are 2 ways to get one 6 and the other 4)

24. Two or More Random Variables Frequency of applications during a given week

25. Two or More Random Variables Joint Probability Distribution

26. Two or More Random Variables Joint Probability Distribution Marginal Probabilities

27. Computing Joint Probability The Probability of a Joint Event, A and B: P(A and B) Number of Event Outcomes from both A and B = Total Number of Possible Outcomes in Sample Space e.g. P(Red CardandAce) =

28. Joint Probability Using Contingency Table Event Total B1 B2 Event A1 P(A1andB1) P(A1andB2) P(A1) P(A2andB2) A2 P(A2) P(A2 and B1) 1 Total P(B1) P(B2) Marginal (Simple) Probability Joint Probability

29. Computing Compound Probability The Probability of a Compound Event, A or B: e.g. P(Red Card orAce)

30. Compound ProbabilityAddition Rule P(A1orB1 ) = P(A1) +P(B1) - P(A1 andB1) Event Total B1 B2 Event A1 P(A1and B1) P(A1and B2) P(A1) P(A2 andB1) A2 P(A2 and B2) P(A2) 1 Total P(B1) P(B2) For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

31. Computing Conditional Probability The Probability of Event Agiven thatEvent B has occurred: P(A B) = e.g. P(Red Cardgiven that it isanAce) =

32. Conditional Probability Using Contingency Table Conditional Event: Draw 1 Card. Note Kind & Color Color Type Total Red Black Revised Sample Space 2 2 4 Ace 24 24 48 Non-Ace 26 26 52 Total

33. Conditional Probability and Statistical Independence Conditional Probability: P(AB) = Multiplication Rule: P(A and B) = P(AB) • P(B) = P(BA) • P(A)

34. Conditional Probability and Statistical Independence (continued) Events are Independent: P(AB) = P(A) Or, P(BA) = P(B) Or, P(A and B) = P(A) • P(B) Events A and B are Independent when the probability of one event, A is not affected by another event, B.

35. Discrete Probability Distribution Example Event: Toss 2 Coins. Count # Tails. • Probability distribution • Valuesprobability • 0 1/4 = .25 • 1 2/4 = .50 • 2 1/4 = .25 T T T T

36. Discrete Random Variable Summary Measures • Expected value (The mean) • Weighted average of the probability distribution •  = E(X)= xi p(xi) • In slow-moving product demand example, • the expected value is : • E(X)= 0  0.1 + 1 .2 + 2  .4 + 3  .3 = 1.9 The average demand on the long run is 1.9

37. Discrete Random Variable Summary Measures • Variance • Weighted average squared deviation about mean • Var[X] =  = E [ (xi -E(X) )2]= (xi - E(X) )2p(xi) • For the Product demand example, the variance is: • Var[X] =  = (0 – 1.9)2(.1) + (1 – 1.9)2(.2) + • (2 – 1.9)2(.4) + (3 – 1.9)2(.3) = .89

38. Important Discrete Probability Distribution Models Discrete Probability Distributions Binomial Poisson

39. Bernoulli Distribution • Two possible mutually exclusive outcomes with constant probabilities of occurrence • “Success” (x=1) or “failure” (x=0) • Example : Response to telemarketing • The probability mass function is • p(x) = p if x=1 • P(x) = 1- p if x=0 Where p is the probability of success

40. Binomial Distribution • ‘N’ identical trials • Example: 15 tosses of a coin, 10 light bulbs taken from a warehouse • 2 mutually exclusive outcomes on each trial • Example: Heads or tails in each toss of a coin, defective or not defective light bulbs

41. Binomial Distributions • Constant Probability for each Trial • Example: Probability of getting a tail is the same each time we toss the coin and each light bulb has the same probability of being defective • 2 Sampling Methods: • Infinite Population Without Replacement • Finite Population With Replacement • Trials are Independent: • The Outcome of One Trial Does Not Affect the Outcome of Another

42. Binomial Probability Distribution Function n ! X  X n P(X)  1 ) p (  p X ! (  ) ! n X P(X) = probability that Xsuccesses given a knowledge of n and p X= number of ‘successes’ insample, (X = 0, 1, 2, ...,n) p = probability of each ‘success’ n = sample size Tails in 2 Tosses of Coin XP(X) 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25

43. Binomial Distribution Characteristics n = 5p = 0.1 P(X) Mean .6  E ( X )   np .4 .2 e.g. = 5 (.1) = .5 0 X 0 1 2 3 4 5 Standard Deviation n = 5p = 0.5 P(X) )    p np ( 1 .6 .4 .2 e.g.=5(.5)(1 - .5) = 1.118 X 0 0 1 2 3 4 5

44. Computing Binomial Probabilities using Excel Function BINOMDIST

45. P ( X  x |  -  x e  x ! Poisson Distribution • Poisson process: • Discrete events in an ‘interval’ • The probability ofone successin an interval isstable • The probability of more than one success in this interval is 0 • Probability of success is • Independent from interval to • Interval • Examples: • # Customers arriving in 15 min • # Defects per case of light bulbs

46. Poisson Distribution Function  X e  P ( X )  X ! P(X) = probability ofXsuccesses given = expected (mean) number of ‘successes’ e = 2.71828 (base of natural logs) X= number of ‘successes’per unit e.g. Find the probability of4customers arriving in 3 minutes when the mean is3.6 -3.6 4 e 3.6 P(X) = =.1912 4!

47. Poisson Distribution Characteristics Mean = 0.5 P(X) .6   E ( X )   .4 N .2   0 X X P ( X ) i i 0 1 2 3 4 5 i  1 = 6 P(X) .6 Standard Deviation .4 .2    0 X 0 2 4 6 8 10

48. Computing Poisson Probabilities using Excel Function POISSON

49. Continuous Probability Distributions • Uniform • Triangular • Normal • Exponential

50. The Uniform Distribution • Equally Likely chances of occurrences of RV values between a maximum and a minimum • Mean = (b+a)/2 • Variance = (b-a)2/12 • ‘a’ is a location parameter • ‘b-a’ is a scale parameter • no shape parameter f(x) 1/(b-a) a b x