Probability Distributions for Discrete Variables: Probability Distributions for Discrete Variables Farrokh Alemi Ph.D .
Discrete Probability Distributions: Bernoulli Geometric Binomial Poisson Discrete Probability Distributions
Definitions: Density function Definitions
Definitions: Cumulative probability function Definitions
Definitions: Definitions
Expected Value: Expected Value Expected Value for variable x
Expected Value: Expected Value Probability of event “i”
Expected Value: Expected Value Value of event “i”
Expected Value: Expected Value Summed over all possible events
Example: Example
Example: Example
Example: Example Expected medication errors
Example: Example
Example: Example
Example: Example
Density & Cumulative Distributions: Density & Cumulative Distributions
Typical Probability Density Functions: Bernoulli Binomial Geometric Poisson Typical Probability Density Functions
Bernoulli Probability Density Function: Mutually exclusive Bernoulli Probability Density Function
Bernoulli Probability Density Function: Exhaustive Bernoulli Probability Density Function
Bernoulli Probability Density Function: Bernoulli Probability Density Function
Bernoulli Probability Density Function: Bernoulli Probability Density Function
Independent Bernoulli Trials: Independence = H istory does not matter Independent Bernoulli Trials
Independent Bernoulli Trials: Independent Bernoulli Trials Patient elopes No event Patient elopes No event Patient elopes No event Month 1 Month 2 Month 3
Geometric Density Function: Geometric Density Function K-1 non-occurrence of the event occurrence of the event
Geometric Density Function: Geometric Density Function
Geometric Density Function: Geometric Density Function
Do One: No medication errors have occurred in the past 90 days. What is the maximum daily probability of medication error in our facility ? Do One
Do One: The time between patient falls was calculated to be 3 days, 60 days and 15 days. What is the daily probability of patient falls? Do One
Binomial Probability Distribution: Number of k occurrences of the event in n independent trials Binomial Probability Distribution
Independent Bernoulli Trials: Independent Bernoulli Trials
Independent Bernoulli Trials: Independent Bernoulli Trials Month
Independent Bernoulli Trials: Independent Bernoulli Trials Month (1-P) x P x P
Independent Bernoulli Trials: Independent Bernoulli Trials Month (1-P) x P x P
Independent Bernoulli Trials: Independent Bernoulli Trials P x (1-P) x P
Binominal Probability Distribution: Different combinations Success probabilities Failure probabilities Binominal Probability Distribution
Binomial Probability Distribution: Binomial Probability Distribution n! is n factorial and is calculated as 1*2*3*…*n Possible ways of getting k occurrences in n trials
Binomial Probability Distribution: Binomial Probability Distribution k occurrences of the event
Binomial Probability Distribution: Binomial Probability Distribution n-k non-occurrence of the event
6 Trials of Binomial p=1/2: 6 Trials of Binomial p=1/2
6 Trials of Binomial p=1/2: 6 Trials of Binomial p=1/2 The expected value of a Binomial distribution is np . The variance is np (1-p)
6 Trials of Binomial p=0.05: 6 Trials of Binomial p=0.05
Example: If the monthly probability of elopement is 0.05, how many patients will elope in 2 years? Example
Example: If the monthly probability of elopement is 0.05, how many patients will elope in 2 years? Example
Example: 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? Example
Example: 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? Example
Do One: Which is more likely, 2 patients failing to comply with medication orders in 15 days or 3 patients failing to comply with medication orders in 30 days. Do One
Poisson Density Function: Large number of trials Small probabilities of occurrence Poisson Density Function
Poisson Density Function: Poisson Density Function Expected number of trials, np
Poisson Density Function: Poisson Density Function k is number of sentinel occurrences
Take Home Lesson: Repeated independent Bernoulli trials is the foundation of many distributions Take Home Lesson
Do One: What is the probability of observing one or more security violations, when the daily probability of violations is .01 and we are monitoring the organization for 4 months. Do One
Do Another: 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 ? Do Another