Loading in 5 sec....

Probability Distributions for Discrete VariablesPowerPoint Presentation

Probability Distributions for Discrete Variables

- By
**mika** - Follow User

- 62 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Probability Distributions for Discrete Variables' - mika

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Probability Distributions for Discrete Variables

### Take Home Lesson

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 [email protected]

Lecture Outline

- What is probability?
- Discrete Probability Distributions
- Assessment of rare probabilities
- Conditional independence
- Causal modeling
- Case based learning
- Validation of risk models
- Examples

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

Definitions

- Function
- Density function
- Distribution function

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”

Calculation of Expected Value from Density Function

Expected medication errors

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

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.

Typical Probability Density Functions

- Bernoulli
- Binomial
- Geometric
- Poisson

Bernoulli Probability Density Function

- Mutually exclusive
- Exhaustive
- Occurs with probability of p

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?

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

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

Geometric Probability Density Function

- Expected number of trials prior to occurrence of the event

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?

Binomial Probability Distribution

- Independent repeated Bernoulli trials
- Number of k occurrences of the event in n trials

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

k occurrences of the even

Possible ways of getting k occurrences in n trials

Binomial Probability Distribution

k occurrences of the even

n-k non-occurrence of the event

Possible ways of getting k occurrences in n trials

Binomial Density Function for 6 Trials, p=1/2

The expected value of a Binomial distribution is np. The variance is np(1-p)

Exercise

- If the daily probability of elopement is 0.05, how many patients will elope in a year?

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?

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?

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.

Poisson Density Function

- Approximates Binomial distribution
- Large number of trials
- Small probabilities of occurrence

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

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?

Repeated independent Bernoulli trials is the foundation of many distributions

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?

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?

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?

Download Presentation

Connecting to Server..