DISCRETE PROBABILITY
DISTRIBUTIONS
Farrokh Alemi Ph.D.
Discrete Probability Distributions
Bernoulli
Geometric
Binomial
Poisson
Definitions
Density function
Definitions
Cumulative probability function
Definitions
Expected Value
Expected Value
Expected Value
Expected Value
Example
Example
Example
Example
Probability
density
Events
function
0 medication errors
0.74
1 medication error
2 medication errors
Otherwise
Cumulative
distribution
function
0.74
0.22
0.04
0.96
1
0
1
Example
Example
Density & Cumulative Distributions
Typical Probability Density Functions
Bernoulli
Geometric
Binomial
Poisson
Bernoulli Probability Density Function
Mutually exclusive
Bernoulli Probability Density Function
Exhaustive
Bernoulli Probability Density Function
Events
Does not occur
Occurs
Probability
density
function
1-P
p
Bernoulli Probability Density Function
Independent Bernoulli Trials
Independent Bernoulli Trials
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
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
Number of k occurrences of the event in n
independent trials
Independent Bernoulli Trials
Independent Bernoulli Trials
Independent Bernoulli Trials
Independent Bernoulli Trials
P x (1-P) x P
Binominal Probability Distribution
Different combinations
Success probabilities
Failure probabilities
Binomial Probability Distribution
Binomial Probability Distribution
Binomial Probability Distribution
6 Trials of Binomial p=1/2
6 Trials of Binomial p=1/2
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
If the monthly probability of elopement is 0.05,
how many patients will elope in 2 years?
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
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?
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.
Poisson Density Function
Large number of trials
Small probabilities of occurrence
Poisson Density Function
k
Poisson Density Function
k
Take Home Lesson
Repeated independent Bernoulli trials is the
foundation of many distributions
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 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?