Discrete Probability Distributions

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This is part of open online course on managerial statistics. This lecture is on discrete distributions. The complete course and other videos are available at http://openonlinecourses.com/statistics

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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