Transcript Chapter 5

Chapter 5
Discrete Probability Distributions
Probability Experiment
A probability experiment is any activity that
produces uncertain or “random” outcomes
Random Variable
A random variable is a rule or function that
translates the outcomes of a probability
experiment into numbers.
Table 5.1 Illustrations of Random
Variables
EXPERIMENT
Possible Random Variables
Type of Variable
Commuting to work
Time it takes to get to work
Number of red lights on the way
Amount of gas consumed
Continuous
Discrete
Continuous
Advertising a product
Number of customer responses
Number of units sold
Discrete
Discrete
Taking inventory
Number of damaged items found
Remaining shelf life of an item
Discrete
Continuous
Playing a round of golf
Driving distance off the first tee
Number of pars
Number of lost balls
Continuous
Discrete
Discrete
Manufacturing a product
Amount of waste produced (lbs.)
Number of units finished in an hour
Continuous
Discrete
Interviewing for a job
Number of rejections
Duration of the interview
Elapsed time before being hired
Discrete
Continuous
Continuous
Buying stocks
Number of your stocks that increase
in value
Amount of sleep lost from worry
Discrete
Continuous
Discrete Random Variable
A discrete random variable has separate
and distinct values, with no values
possible in between.
Continuous Random Variable
A continuous random variable can take on
any value over a given range or interval.
Probability Distribution
A probability distribution identifies the
probabilities that are assigned to all
possible values of a random variable.
Producing a Discrete Probability Distribution
Step 1: Defining the Random Variable
Step 2: Identifying Values for the Random
Variable
Step 3: Assigning Probabilities to Values
of the Random Variable
Figure 5.1
Probability Tree for the
Management Training Example
J (.7)
Outcome
(1)
Jones Passes
S∩J
x
P(x)
2
.63
S (.9)
Smith Passes
J' (.3)
(2)
S∩J’
1
.27
(3)
S'∩J
1
.07
0
.03
Jones Fails
J (.7)
Jones Passes
S' (.1)
Smith Fails
J' (.3)
(4)
Jones Fails
S'∩J’
Probability Distribution for the Training
Course Illustration
Number of
Managers
Passing
x
Probability
P(x)
0
.03
1
.34
2
.63
1.0
Figure 5.2 Graphing the Management
Training Distribution
P(x)
.6
.3
.1
0
1
2
Number of Managers Passing
x
Expected Value for a
(5.1)
Discrete Probability Distribution
E(x) =  x  P(x)
Distribution Variance
s2
=
 ( x  E( x))
2
 P( x)
(5.2)
The Binomial Distribution
Figure 5.3
Probability Tree for the Coin
Toss Example
H (.4)
H (.4)
3 Heads
.064
2 Heads
.096
T (.6)
H (.4)
H (.4)
2 Heads .096
T (.6)
T (.6)
1 Head
.144
2 Heads
.096
1 Head
.144
1 Head
.144
0 Heads
.216
H (.4)
H (.4)
T (.6)
T (.6)
H (.4)
T (.6)
T (.6)
The Binomial Conditions
(1)
The experiment involves a number of “trials”— that is, repetitions of
the same act. We’ll use n to designate the number of trials.
(2)
Only two outcomes are possible on each of the trials. This is the “bi”
part of “binomial.” We’ll typically label one of the outcomes a success, the
other a failure.
(3)
The trials are statistically independent. Whatever happens on one trial
won’t influence what happens on the next.
(4)
The probability of success on any one trial remains constant
throughout the experiment. For example, if the coin in a coin-toss
experiment has a 40% chance of turning up heads on the first toss, then
that 40% probability must hold for every subsequent toss. The coin can’t
change character during the experiment. We’ll normally use p to represent
this probability of success.
The Binomial Probability Function
P (x) =
(5.4)
n!
x
(n  x)
p (1  p)
(n  x)!x!
Expected Value for a
Binomial Distribution
E(x) = nּ p
(5.5)
Variance for a Binomial Distribution
s2 = nּpּ(1-p)
(5.6)
Standard Deviation for a
Binomial Distribution
s=
n  p  (1  p)
(5.7)
Figure 5.4
P(x)
Some Possible Shapes
for a Binomial Distribution
P(x)
Positively Skewed
0
1
2
3
4
5
6
Symmetric
0
x
1
2
3
P(x)
Negatively Skewed
0
1
2
3
4
5
6
x
4
5
6
x
The Poisson Distribution
The Poisson Conditions
(1) We need to be assessing probability for the number of
occurrences of some event per unit time, space, or
distance.
(2) The average number of occurrences per unit of time,
space, or distance is constant and proportionate to the
size of the unit of time, space or distance involved.
(3) Individual occurrences of the event are random and
statistically independent.
Poisson Probability Function
P(x) =


x
e x!
(5.8)
Figure 5.5
Graphing the Poisson
Distribution for  = 1
P(x)
.3
.2
.1
0
1
2
3
4
mean =  =1
5
6
x
Figure 5.6
Matching Binomial and
Poisson Distributions
P(x)
BINOMIAL
n = 20, p = .10
.3
.2
.1
0
1
2
3
4
5
6
x
P(x)
POISSON
=2
.3
.2
.1
0
1
2
3
4
5
6
x