Probability Distributions for Discrete Variables

1. Probability Distributions for Discrete Variables Farrokh Alemi Ph.D.Professor of Health Administration and PolicyCollege of Health and Human Services, George Mason University4400 University Drive, Fairfax, Virginia 22030703 993 1929 falemi@gmu.edu

2. Lecture Outline • What is probability? • Discrete Probability Distributions • Assessment of rare probabilities • Conditional independence • Causal modeling • Case based learning • Validation of risk models • Examples

3. Lecture Outline • What is probability? • Discrete Probability Distributions • Bernoulli • Geometric • Binomial • Poisson • Assessment of rare probabilities • Conditional independence • Causal modeling • Case based learning • Validation of risk models • Examples

4. Definitions • Function • Density function • Distribution function

5. Definitions

6. Expected Value • Probability density function can be used to calculate expected value for an uncertain event. Summed over all possible events Expected Value for variable X Value of event “i” Probability of event “i”

7. Calculation of Expected Value from Density Function

8. Calculation of Expected Value from Density Function

9. Calculation of Expected Value from Density Function Expected medication errors

10. Exercise • Chart the density and distribution functions of the following data for patients with specific number of medication errors & calculate expected number of medication errors

11. Probability Density & Cumulative Distribution Functions

12. Exercise • If the chances of medication errors among our patients is 1 in 250, how many medication errors will occur over 7500 patients? Show the density and cumulative probability functions.

13. Typical Probability Density Functions • Bernoulli • Binomial • Geometric • Poisson

14. Bernoulli Probability Density Function • Mutually exclusive • Exhaustive • Occurs with probability of p

15. Exercise • If a nursing home takes care of 350 patients, how many patients will elope in a day if the daily probability of elopement is 0.05?

16. Day 1 Day 2 Day 3 Patient elopes Patient elopes Patient elopes No event No event No event Independent Repeated Bernoulli Trials • Independence means that the probability of occurrence does not change based on what has happened in the previous day

17. Geometric Probability Density Function • Number of trials till first occurrence of a repeating independent Bernoulli event K-1 non-occurrence of the event occurrence of the event

18. Geometric Probability Density Function • Expected number of trials prior to occurrence of the event

19. Exercise • No medication errors have occurred in the past 90 days. What is the daily probability of medication error in our facility? • The time between patient falls was calculated to be 3 days, 60 days and 15 days. What is the daily probability of patient falls?

20. Binomial Probability Distribution • Independent repeated Bernoulli trials • Number of k occurrences of the event in n trials

21. Repeated Independent Bernoulli Trials

22. Binomial Probability Distribution n! is n factorial and is calculated as 1*2*3*…*n Possible ways of getting k occurrences in n trials

23. Binomial Probability Distribution k occurrences of the even Possible ways of getting k occurrences in n trials

24. Binomial Probability Distribution k occurrences of the even n-k non-occurrence of the event Possible ways of getting k occurrences in n trials

25. Binomial Density Function for 6 Trials, p=1/2 The expected value of a Binomial distribution is np. The variance is np(1-p)

26. Binomial Density Function for 6 Trials, p=0.05

27. Exercise • If the daily probability of elopement is 0.05, how many patients will elope in a year?

28. Exercise • If the daily probability of death due to injury from a ventilation machine is 0.002, what is the probability of having 1 or more deaths in 30 days? What is the probability of 1 or more deaths in 4 months?

29. Exercise • If the daily probability of death due to injury from a ventilation machine is 0.002, what is the probability of having 1 or more deaths in 30 days? What is the probability of 1 or more deaths in 4 months?

30. Exercise • Which is more likely, 2 patients failing to comply with medication orders in 15 days or 4 patients failing to comply with medication orders in 30 days.

31. Poisson Density Function • Approximates Binomial distribution • Large number of trials • Small probabilities of occurrence

32. Poisson Density Function Λ is the expected number of trials = n p k is the number of occurrences of the sentinel event e = 2.71828, the base of natural logarithms

33. Exercise • What is the probability of observing one or more security violations. when the daily probability of violations is 5% and we are monitoring the organization for 4 months • What is the probability of observing exactly 3 violations in this period?

34. Take Home Lesson Repeated independent Bernoulli trials is the foundation of many distributions

35. Exercise • What is the daily probability of relapse into poor eating habits when the patient has not followed her diet on January 1st, May 30th and June 7th? • What is the daily probability of security violations when there has not been a security violation for 6 months?

36. Exercise • How many visits will it take to have at least one medication error if the estimated probability of medication error in a visit is 0.03? • If viruses infect computers at a rate of 1 every 10 days, what is the probability of having 2 computers infected in 10 days?

37. Exercise • Assess the probability of a sentinel event by interviewing a peer student. Assess the time to sentinel event by interviewing the same person. Are the two responses consistent?