Random Variables (Motivation) and its types

Random Variables and its types

1. Random Variables (Motivation)

2. Random Variables

3. Discrete Random Variables If the set of all possible values of the random variable is either finite or countably infinite , then the random variable is discrete. A discrete random variable is an rv whose possible values either constitute a finite set or else can be listed in an infinite sequence in which there is a first element, a second element, and so on.

4. Continuous Random Variables If the set of all possible values of the random variable is an interval , then the random variable is continuous. A random variable is continuous if its set of possible values consists of an entire interval on the number line. No matter how small the range, the number of possible values for X is always ( uncountably ) infinite

5. Common Distributions Common discrete distributions: • generic discrete • Bernoulli • binomial • geometric • Poisson • Common continuous distributions: • uniform • step • triangular • exponential • normal • lognormal • gamma • Student t • F • beta • Weibull

6. REVIEW OF PROBABILITY THEORY Probability is the measure of the likeliness that an event will occur.

7. Probability Mass Function (Discrete)

8. Probability Mass Function (Discrete) Pmf describes how the total probability mass of 1 is distributed at various points along the axis of possible values of the random variable (where and how much mass at each x).

9. Cumulative Distribution Function Die Example

10. Cumulative Distribution Function

11. Probability Density Function (Continuous)

12. Probability Density Function (Continuous)

13. Probability Density Function (Continuous)

14. Probability Density Function (Continuous)

15. PDF Continued

16. Cumulative Distribution Function

17. Sample Derivation: Uniform CDF

18. Inverse CDF

19. Inverse CDF

20. Inverse CDF

21. Sampling from a Probability Distribution

22. Exponential Distribution A random variable, , with an exponential distribution is widely used to model “random” arrivals in continuous time, especially when the inter-arrival times are iid (e.g., customer inter-arrivals, times to failure, etc.) λ = arrival rate (0.2 emergency calls are arriving in one hour). X~ Expo ( λ ) The state space is S = {0, ∞] µ = 1/ λ , (average response time of an ambulance is equal to 5 hours). variance = 1/ λ² A sequence or other collection of random variables is independent and identically distributed ( iid ) if each random variable has the same probability distribution as the others and all are mutually independent

23. Sample Derivation: Exponential CDF X is sample and F(x) is rand function in excel

24. Normal Distribution CDF * If cumulative is TRUE, NORMDIST returns the CDF; if FALSE, it returns the pmf .

25. Example: Exponential Distribution

26. Exponential Distribution: Spreadsheet Simulation

27. Exponential Distribution: Histogram of RAND generated by excel

28. Exponential Distribution: Histogram of generated samples

29. Example: Normal Distribution

30. Normal Distribution: Spreadsheet Simulation

31. Normal Distribution: Histogram of generated samples

32. BETA Distribution: Overview

33. BETA Distribution: Overview

34. BETA Distribution: Some Formulas

35. BETA Distribution: Spreadsheet Simulation

36. Triangular Distribution Alternative for Beta distribution. Parameters of Beta distribution are not easily estimated and does not have a closed form for its cdf . Can be used even when data is limited. It is also called “lack of Knowledge” distribution. Widely used in Risk Analysis of disasters Project Management to model events which take place within an interval defined by a minimum and maximum value. Reference: The triangular distribution as a proxy for the beta distribution in risk analysis By DAVID JOHNSON, The Statistician (1997) 46, No. 3, pp. 387-398.

37. Triangular Distribution A random variable, , with a triangular distribution is used when it is assumed that the likelihood of values first increases linearly to some maximal value (mode ), and then decreases linearly

38. Triangular Distribution Triangular distribution can be expected to produce a distribution function, which is very similar to that of the beta distribution

39. Triangular Distribution

40. Triangular Distribution

41. Triangular Distribution The pdf is as follows:

42. Triangular Distribution The cdf is given by: The inverse cdf is as follows:

43. The Central Limit Theorem (CLT)

44. The Central Limit Theorem (CLT)

45. The Central Limit Theorem (CLT) Reference: Exploring Research by Neil Salkind , seventh edition

46. The Central Limit Theorem (CLT)

47. Simulation Example Arrival Rate= 1/ inter arrival time For example; If 12 customers enter a store per hour. Arrival rate is 12 customers per hour. The time between each arrival is; = 1/12= 0.083 hours= 5 minutes * The inter arrival time is the time between each arrival into the system arrival is 5 minutes.

48. Simulation Example

49. Exponential Distribution Rescue 211 Ambulance Example λ = arrival rate (0.2 emergency calls are arriving in one hour) µ = 1/ λ , (average response time of an ambulance is equal to 5 hours)