Probability Distributions for Discrete Variables

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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_at_gmu.edu

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Lecture Outline

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

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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

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Definitions

• Function
• Density function
• Distribution function

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Definitions
Events Probability density function Cumulative distribution function
0 medication errors 0.90 0.90
1 medication error 0.06 0.96
2 medication errors 0.04 1
Otherwise 0 1
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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
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Calculation of Expected Value from Density
Function
Events Probability density function Value times probability
0 medication errors 0.90 0(0.90)0
1 medication error 0.06
2 medication errors 0.04
Otherwise 0

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Calculation of Expected Value from Density
Function
Events Probability density function Value times probability
0 medication errors 0.90 0(0.90)0
1 medication error 0.06 0.06
2 medication errors 0.04 0.08
Otherwise 0 0

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Calculation of Expected Value from Density
Function
Events Probability density function Value times probability
0 medication errors 0.90 0
1 medication error 0.06 0.06
2 medication errors 0.04 0.08
Otherwise 0 0
Total 0.12
Expected medication errors
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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

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Probability Density Cumulative Distribution
Functions
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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.

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Typical Probability Density Functions

• Bernoulli
• Binomial
• Geometric
• Poisson

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Bernoulli Probability Density Function

• Mutually exclusive
• Exhaustive
• Occurs with probability of p

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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?

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Independent Repeated Bernoulli Trials

• Independence means that the probability of
  occurrence does not change based on what has
  happened in the previous day

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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
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Geometric Probability Density Function

• Expected number of trials prior to occurrence of
  the event

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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?

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Binomial Probability Distribution

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

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Repeated Independent Bernoulli Trials
Probability of exactly two elopement in 3 days
On day 1 and 2 not 3 p p (1-p)
On day 1 not 2 and 3 p (1-p) p
On day 2 3 and not 1 P p (1-p)
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Binomial Probability Distribution
n! is n factorial and is calculated as 123n
Possible ways of getting k occurrences in n trials
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Binomial Probability Distribution
k occurrences of the even
Possible ways of getting k occurrences in n trials
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Binomial Probability Distribution
k occurrences of the even
n-k non-occurrence of the event
Possible ways of getting k occurrences in n trials
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Binomial Density Function for 6 Trials, p1/2
The expected value of a Binomial distribution is
np. The variance is np(1-p)
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Binomial Density Function for 6 Trials, p0.05
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Exercise

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

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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?

Number of trials 30
Daily probability 0.002
Number of deaths 0
Probability of 0 deaths 0.942
Probability of 1 or more deaths 0.058
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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?

Number of trials 30
Daily probability 0.002
Number of deaths 0
Probability of 0 deaths 0.942
Probability of 1 or more deaths 0.058
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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.

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Poisson Density Function

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

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Poisson Density Function
? 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
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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?

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Take Home Lesson

• Repeated independent Bernoulli trials is the
  foundation of many distributions

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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?

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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?

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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?