Discrete Probability Distributions
Discrete vs. Continuous
• Discrete
▫ A random variable (RV) that can take only certain
values along an interval:
 Cars passing by a point
 Results of coin toss
 Students taking a class
• Continuous
▫ An RV that can take on any value at any point
along an interval:
 Temperature, time, distance, money etc.
Frequency Distribution
• Number of times an observation occurs in a
given population.
What is a variable?
• A symbol (A, B, x, y, etc.) that can take on any of
a specific set of values
▫ X=number of heads
▫ Y= temperature
• Random variable
▫ The outcome of a statistical experiment
Random variable notation
• Capital letter represents the RV
▫ X=total number of heads in 4 tosses
▫ P(X) represents the probability of X
• Lower-case letter represents one of the values of
the RV
▫ P(X=x) is the probability the RV will assume a
specific value
▫ P(X=2) is the probability that we will have exactly
2 heads in the 4 tosses
Probability Distribution
• Relative frequency distribution that should,
theoretically, occur for observations from a
given population.
Outcome
HH
HT
TH
TT
X
0
1
2
#heads
2
1
1
0
Probability
P(X)
.25
0.25
.25
0.50
.25
0.25
.25
Cumulative probability distribution
• Probability the value of a RV falls within a
specified range.
• Coin toss: P(X≤1)
# Heads
P(X=x)
P(X≤x)
0
0.25
0.25
1
0.50
0.75
2
0.25
1.00
Characteristics of a Discrete
Probability Distribution
• For any value of x
• The values of x are exhaustive, i.e. the
distribution contains all the possible values
• The values of x are mutually exclusive; i.e., only
one value can occur for an experiment
• The sum of the probabilities equals 1
Mean and standard deviation
• Mean of discrete distribution is called expected
value
• Variance
• Standard Deviation
Practice
• Determine the Mean (µ) or Expected Value
(E(x)) for the following data.
X
0
1
2
P(x)
0.6
0.3
0.1
Practice
• A music shop is holding a promotion in which
the customer rolls a die and deducts a dollar
from the price of a CD equal to the number that
he rolls.
• If the owner pays $5.00 for each disk and prices
them at $9.00, what will his expected profit be
on each CD during this promotion?
Binomial Distributions
• There are 2 or more identical trials
• In each trial, there can be only 2 outcomes
(success or failure)
• Trials are statistically independent
▫ Outcome of one trial does not influence outcome
of the next
• Probability of success remains the same from
one trial to the next
Binomial Experiment?
• An article in a 1988 issue of The New England
Journal of Medicine talked about a TB outbreak.
▫ One person caught the disease in 1995
▫ 232 workers sampled from a very large population
were given a TB test
▫ The number of workers testing positive is the
variable of interest
• If we test all 232 workers for the disease, is this a
binomial experiment?
Binomial Experiment?
• Bill has to sell 3 cars to meet his monthly quota.
He has 5 customers, but 3 of them are interested
in the same car and will leave if that car is sold.
• He has a 30% chance of a sale with each
customer.
• Is this a binomial experiment?
Binomial Distributions
• Probability of exactly x successes in n trials:
• Where:
▫
▫
▫
▫
π = probability of success for any trial
n = number of trials
x = number of successes
(1-x) = number of failures
Binomial Distributions
• Expected value
• Variance
Binomial Distributions in Excel
• =binom.dist(number_s,trials,probability_s,cumulative)
• Where:
▫
▫
▫
▫
Number_s = number of successes
Trials
Probability_s = probability of success
Cumulative:
 False, if we want the probability of x
 True, if we want the probability of all the variables up to and
including x
• Example, in the previous problem
▫ P(X=5)
▫ =binom.dist(5,5,.1,false)
Binomial Experiment
• We’re going to select 5 households at random in
a city where the unemployment rate is 10% to
see if the head of the household is unemployed.
What is the probability that all 5 are employed?
• Is this a binomial experiment?
▫ Why or why not?
These examples came from the StatTrek website: http://stattrek.com/Lesson1/Statistics-Intro.aspx?Tutorial=Stat
Acceptance to college
• The probability that a student is accepted to a
prestigious college is 0.3. If 5 students apply,
what is the probability that at most 2 are
accepted?
Probability Distribution
0.4
0.35
Probability
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
# Accepted
3
4
5
Cumulative Probability Distribution
1
0.9
0.8
Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
# Accepted
3
4
5
Coin flipping - again
• What is the probability of getting 45 or fewer
heads in 100 tosses of a fair coin?
Probability Distribution
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
60
70
80
90
100
Cumulative Probability Distribution
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
The World Series
• What is the probability that the World Series will
last 4 games?
• 5 games?
• 6 games?
• 7 games?
▫ Assume the teams are evenly matched.
Poisson Distribution
• Applies for events occurring over time, space, or
distance
• Examples:
▫
▫
▫
▫
Number of cars driving past a point
Number of defects per foot in manufactured pipe
Number of knots in a section of wood panel
Number of accidents per day at a job site
Poisson Distribution
e is the base of the natural logarithm system
and is equal to 2.71828
Any number raised to a negative exponent is
the same as 1 divided by that number raised to
its exponent. Example: 2-2 is the same as 1/22
Poisson Distribution
• There were 438 children born in a small town
last year.
• What is the probability that, on any given day,
no children were born?
Poisson Distribution in Excel
• =poisson.dist(x,mean,cumulative)
• Where:
▫ X=the number we’re looking for
▫ Mean = lambda
▫ Cumulative
 True = probability of all values up to and including x
 False = probability of x
• =poisson.dist(0,1.2,false)
Probability Distribution
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Cumulative Poisson Distribution
• Suppose the average number of lions seen on a
1-day safari is 5. What is the probability that
tourists will see fewer than four lions on the next
1-day safari?
Cumulative Poisson Distribution
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Hypergeometric Distribution
• Sampling without replacement
• Compare to binomial
▫ There are 2 or more identical trials
▫ In each trial, there can be only 2 outcomes
(success or failure)
▫ Trials are statistically independent
▫ Probability
of not
success
remainsindependent
the same from one
Trials are
statistically
trial to the next
Probability of success changes from one
▫ The random variable is the number of successes in
trial to the next
n trials
Hypergeometric Distribution
Where:
N=size of the population
n=size of the sample
s=number of successes in the population
x=number of successes in the sample
Hypergeometric in Excel
• =hypgeom.dist(sample_s, number_sample,
population_s, number_population, cumulative)
• Where:
sample_s=number of successes in the sample
number_sample=size of the sample
population_s=number of successes in the
population
number_population=size of the population
cumulative=same as before
• =hypgeom.dist(2, 4, 6, 20, false)
Hypergeometric Distribution
• 20 businesses filed tax returns
• 6 of the returns were filled out incorrectly
• The IRS has randomly selected 4 of the 20
returns to audit
• What is the probability that exactly 2 of the 4
selected for audit will be filled out incorrectly?
Hypergeometric Distribution
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
Cumulative Hypergeometric
• Suppose we select 5 cards from an ordinary deck
of playing cards. What is the probability of
obtaining 2 or fewer hearts?
Cumulative Hypergeometric
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Summary
• Random variables
▫ Discrete v. continuous
• Probability distributions
▫ Cumulative distributions
•
•
•
•
Expected values
Binomial
Poisson
Hypergeometric