Download Chapter 5

Discrete Probability
Distributions
Chapter 5
MSIS 111
Prof. Nick Dedeke
Learning Objectives
Distinguish between discrete random
variables and continuous random variables.
Know how to determine the mean and
variance of a discrete distribution.
Identify the type of statistical experiments
that can be described by the binomial
distribution, and know how to work such
problems.
Learning Objectives -- Continued
Decide when to use the Poisson
distribution in analyzing statistical
experiments, and know how to work
such problems.
Decide when binomial distribution
problems can be approximated by the
Poisson distribution, and know how to
work such problems.
Random Variable
Random Variable -- a variable which
associates a numerical value to the
outcome of a chance experiment
Discrete vs. Continuous Distributions
Discrete Random Variable – arise from
counting experiments. It has a finite number
of possible values or an infinite number of
possible values that can be arranged in sequence



Number of new subscribers to a magazine
Number of bad checks received by a restaurant
Number of absent employees on a given day
Continuous Random Variable – arise from
measuring experiments. It takes on values at
every point over a given interval



Current Ratio of a motorcycle distributorship
Elapsed time between arrivals of bank customers
Percent of the labor force that is unemployed
Discrete and Continuous Distributions
Discrete


Binomial distributions
Poisson distributions
Continuous






Uniform distributions
normal distributions
exponential distributions
T distributions
chi-square distributions
F distributions
Experiment
Experiment: We want to flip a coin
twice. What are all the possibilities
of the experiment?
Experiment
Experiment: We want to flip a coin twice.
What are all the possibilities of the
experiment?
Outcomes possible: HH, HT, TH, TT
To assign numerical variables to the
outcomes, we have to define the random
variables
Let X be number of heads in outcome
Let Y be number of tail in outcome
Deriving Random Variables for
Experiment
Experiment: We want to flip a coin twice. What are all the
possibilities of the experiment?
Outcomes possible: HH, HT, TH, TT
Let X be number of heads in outcome
Let Y be number of tails in outcome
Outcome
HH
HT
TH
TT
values of X
2
1
1
0
values of Y
0
1
1
2
Exercise: Deriving Random Variables
for Experiment
Experiment: We want to flip a coin thrice. What are all
the possibilities of the experiment? What are the
random variables?
Outcomes possible:
Let X be number of heads in outcome
Let Y be number of tails in outcome
Exercise: Deriving Discrete Probability
Distribution
Experiment: We flip a coin thrice. What is the probability
distribution for X random variable the experiment? Let X be the
number of heads in outcome. What is the probability distribution
for Y random variable?
Outcome values of X
HHH
3
HHT
2
HTH
2
HTT
1
THH
2
THT
1
TTH
1
TTT
0
values of Y
0
1
1
2
1
2
2
3
X
0
1
2
3
Fi
1
3
3
1
8
P(x)
0.125
0.375
0.375
0.125
1.00
Exercise: Deriving Discrete Probability
Distribution
Experiment: We investigate the number of people in a gym that injured
themselves before we registered. The data is provided below. What is the
probability distribution for X random variable the experiment? Let X be the
number of muscle injuries in outcome. Let X be the number of bone
injuries in outcome. What is the probability distribution for Y random
variable?
Outcome
1st pool
2nd pool
3rd pool
4th pool
5th pool
6th pool
7th pool
8th pool
values of X
4
2
4
4
3
3
1
0
values of Y
0
1
1
2
1
2
2
0
Example: Discrete Distributions & Graphs
Distribution of Daily
Crises
Number of
Probability
Crises
0
1
2
3
4
5
0.37
0.31
0.18
0.09
0.04
0.01
P
r
o
b
a
b
i
l
i
t
y
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Number of Crises
4
5
Requirements for a
Discrete Probability Function
Probabilities are between 0 and 1,
inclusively
0  P( X )  1 for all X
Total of all probabilities equals 1
 P( X )  1
over all x
Requirements for a Discrete
Probability Function -- Examples
X
P(X)
X
P(X)
X
P(X)
-1
0
1
2
3
.1
.2
.4
.2
.1
1.0
-1
0
1
2
3
-.1
.3
.4
.3
.1
1.0
-1
0
1
2
3
.1
.3
.4
.3
.1
1.2
PROBABILITY
DISTRIBUTION
: YES
NO
NO
Example: Mean of a Discrete Distribution
  E X    X  P( X )
X
-1
0
1
2
3
P(X) X  P( X)
.1
.2
.4
.2
.1
-.1
.0
.4
.4
.3
1.0
 = 1.0
Variance and Standard Deviation
of a Discrete Distribution
  E  X    X  P( X )  1


2
2


P
(
X
)

1
.
2

  X 

  1.2  1.10
2
X
P(X)
X 
-1
0
1
2
3
.1
.2
.4
.2
.1
-2
-1
0
1
2
( X   ) ( X   )  P( X )
2
4
1
0
1
4
2
.4
.2
.0
.2
.4
1.2
Example: Mean of the Crises Data
  E X    X  P( X )  115
.
X
P(X)
X P(X)
0
.37
.00
1
.31
.31
2
.18
.36
3
.09
.27
4
.04
.16
5
.01
.05
P
r
o
b
a
b
i
l
i
t
y
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Number of Crises
1.15
4
5
Example: Variance and Standard
Deviation of Crises Data
    X     P( X )  1.41
2
2
X
P(X)
(X- )
(X- )
0
.37
-1.15
1.32
.49
1
.31
-0.15
0.02
.01
2
.18
0.85
0.72
.13
3
.09
1.85
3.42
.31
4
.04
2.85
8.12
.32
5
.01
3.85
14.82
.15
2
(X- ) 2  P(X)
1.41



2
 141
.  119
.
Exercise: Deriving Discrete Probability
Distribution
Experiment: We investigate the number of people in two gyms that
injured themselves. The data is provided below. What is the
probability distribution for X random variable for each experiment?
What is the mean and standard deviations for each gym? What is
the probability that equal to or greater than 3 injuries occur in
each gym? Let X be the number of injuries in outcome.
X1 P(x1)
4
2
4
4
3
3
1
0
X2
4
2
3
4
3
2
2
1
P(x2)
Exercise: Deriving Discrete Probability
For the data below, find the following probabilities:
What is the probability that greater than 3 injuries occur in the gym?
What is the probability that less than 2 injuries occur in the gym?
What is the probability that greater than or equal to 2 but less than 4
injuries occur in the gym? Let X be the number of injuries in
outcome.
X1
0
1
2
3
4
Fi
1
1
1
2
3
8
P(x1)
0.125
0.125
0.125
0.250
0.375
1.00

X*P(x1) 
0
-2.625
0.125 -1.625
0.250 -0.625
0.750 0.375
1.50
1.375
2.625
X 1 

X 1 

2
6.8906
2.6406
0.3906
0.1406
1.8906
11.953

P ( x1) X 1 
0.0000
2.6406
0.7812
0.4218
7.5624
11.406

2
Examples