Transcript Discrete Random Variables

Discrete Random Variables
Discrete Probability Distributions
•
Binomial Distribution
•
Poisson Distribution
•
Hypergeometric Distribution
Econ10/Mgt 10
Stuffler
1
Discrete Probability Distributions
Discrete Random Variables – Sample Space includes all
mutually exclusive outcomes
Probabilities – from subjective, frequency or subjective
methods
Two conditions apply:
Econ10/Mgt 10
Stuffler
2
Discrete Probability Distributions
Probability distributions can be estimated from relative
frequencies. Consider the discrete (countable) number of
televisions per household (millions) from US survey data:
Econ10/Mgt 10
Stuffler
3
Discrete Probability Distributions
Probability distributions can be estimated from relative
frequencies. Consider the discrete (countable) number of
televisions per household from US survey data:
1,218 ÷ 101,501 = 0.012
Econ10/Mgt 10
Stuffler
4
Discrete Probability Distributions
Probability distributions can be estimated from relative
frequencies. Consider the discrete (countable) number of
televisions per household from US survey data:
EX: P(X=4) = P(4) = 0.076 = 7.6%
Econ10/Mgt 10
Stuffler
5
Discrete Probability Distributions
What is the probability there is at least one television
but no more than three in any given household?
Econ10/Mgt 10
Stuffler
6
Discrete Probability Distributions
What is the probability there is at least one television
but no more than three in any given household?
Econ10/Mgt 10
Stuffler
7
Discrete Probability Distributions
What is the probability there is at least one television
but no more than three in any given household?
“at least one television but no more than three”
P(1 ≤ X ≤ 3) = P(1) + P(2) + P(3) = .319 + .374 + .191 = .884
Econ10/Mgt 10
Stuffler
8
Discrete Probability Distributions
Assume a mutual fund
salesman knows
that there is 20% chance
of closing a sale on each
call he makes.
Econ10/Mgt 10
Stuffler
9
Discrete Probability Distributions
What is the probability distribution of the
number of sales if he plans to call three
customers?
Let S denote success, making a sale:
P(S) = .20,
then Sʹ, not making a sale:
P(Sʹ) = .80
Econ10/Mgt 10
Stuffler
10
Discrete Probability Distributions
• Developing a Probability Distribution Tree
Sales Call 1
P(S)=.2
P(S’)=.8
Econ10/Mgt 10
Stuffler
11
Discrete Probability Distributions
• Developing a Probability Distribution Tree
Sales Call 1
Sales Call 2
P(S)=.2
P(S)=.2
P(S’)=.8
P(S’)=.8
P(S)=.2
P(S’)=.8
Econ10/Mgt 10
Stuffler
12
Discrete Probability Distributions
• Developing a Probability Distribution Tree
Sales Call 1
Sales Call 2
P(S)=.2
P(S)=.2
P(S’)=.8
Sales Call 3
P(S)=.2
SSS
P(S’)=.8
P(S)=.2
S S S’
S S’ S
P(S’)=.8
P(S)=.2
S S’ S’
S’ S S
P(S’)=.8
P(S)=.2
S’ S S’
S’ S’ S
P(S’)=.8
S’ S’ S’
P(S’)=.8
P(S)=.2
P(S’)=.8
Econ10/Mgt 10
Stuffler
13
Discrete Probability Distributions
• Developing a Probability Distribution Tree
Sales Call 1
Sales Call 2
P(S)=.2
P(S)=.2
P(S’)=.8
Sales Call 3
P(S)=.2
SSS
P(S’)=.8
P(S)=.2
S S S’
S S’ S
P(S’)=.8
P(S)=.2
S S’ S’
S’ S S
P(S’)=.8
P(S)=.2
S’ S S’
S’ S’ S
P(S’)=.8
S’ S’ S’
P(S’)=.8
P(S)=.2
P(S’)=.8
Econ10/Mgt 10
Stuffler
14
Discrete Probability Distributions
• Developing a Probability Distribution
Sales Call 1
Sales Call 2
P(S)=.2
P(S)=.2
P(S’)=.8
Sales Call 3
P(S)=.2
SSS
P(S’)=.8
P(S)=.2
S S S’
S S’ S
P(S’)=.8
P(S)=.2
S S’ S’
S’ S S
P(S’)=.8
P(S)=.2
S’ S S’
S’ S’ S
P(S’)=.8
S’ S’ S’
‘)=.8
P(S)=.2
P(S’)=.8
Econ10/Mgt 10
Stuffler
X
3
2
1
0
P(x)
.23 = .008
3(.032)=.096
3(.128)=.384
.83 = .512
15
Discrete Probability Distributions
• Explain how to derive .032 and .128
Sales Call 1
Sales Call 2
P(S)=.2
P(S)=.2
P(SC)=.8
Sales Call 3
P(S)=.2
SSS
P(SC)=.8
P(S)=.2
S S SC
S SC S
P(SC)=.8
P(S)=.2
S SC SC
SC S S
P(SC)=.8
P(S)=.2
SC S SC
SC SC S
P(SC)=.8
SC SC SC
P(SC)=.8
P(S)=.2
P(SC)=.8
Econ10/Mgt 10
Stuffler
X
3
2
1
0
P(x)
.23 = .008
3(.032)=.096
3(.128)=.384
.83 = .512
16
Discrete Probability Distributions
• The P(X=2) is:
Sales Call 1
Sales Call 2
Sales Call 3
(.2)(.2)(.8)= .032
P(S)=.2
P(S)=.2
P(SC)=.8
P(S)=.2
SSS
P(SC)=.8
P(S)=.2
S S SC
S SC S
P(SC)=.8
P(S)=.2
S SC SC
SC S S
P(SC)=.8
P(S)=.2
SC S SC
SC SC S
P(SC)=.8
SC SC SC
P(SC)=.8
P(S)=.2
P(SC)=.8
Econ10/Mgt 10
Stuffler
X
3
2
1
0
P(x)
.23 = .008
3(.032)=.096
3(.128)=.384
.83 = .512
17
Discrete Probability Distributions
A discrete probability distribution
represents a population
Example: Population of number of TVs per household
Example: Population of sales call outcomes
Since we have populations, we can describe
them by computing various parameters:
Population Mean and Population Variance
Econ10/Mgt 10
Stuffler
18
Discrete Probability Distributions
Population Mean (Expected Value)
- Weighted average of all values with the weights
being the probabilities
- Expected value of X, E(X)
Econ10/Mgt 10
Stuffler
19
Discrete Probability Distributions
Population variance - weighted average of the
squared deviations from the mean.
“Short cut” formula for the variance
Standard deviation formula
Econ10/Mgt 10
Stuffler
20
Discrete Probability Distributions
Find the mean, variance, and standard deviation for the
population of the number of color televisions per household.
= 0(.012) + 1(.319) + 2(.374) + 3(.191) + 4(.076) + 5(.028)
= 2.084
Econ10/Mgt 10
Stuffler
21
Discrete Probability Distributions
Find the mean, variance, and standard deviation for the
population of the number of color televisions per household.
= (0 – 2.084)2(.012) + (1 – 2.084)2(.319)+…+(5 – 2.084)2(.028)
= 1.107
Econ10/Mgt 10
Stuffler
22
Discrete Probability Distributions
Find the mean, variance, and standard deviation for the
population of the number of color televisions per household.
= 1.052
Econ10/Mgt 10
Stuffler
23
Discrete Probability Distributions
Special application of Expected Value
Suppose the probability that an insurance agent
makes a sale is .20 and after costs earns a
commission of $525.
If he/she does not make a sale, they must pay $75
in costs.
What is their expected value from a sales call?
Does the benefit exceed the cost or is the
opposite true?
Econ10/Mgt 10
Stuffler
24
Discrete Probability Distributions
Special application of Expected Value
Let X be the discrete random variable of
making a sale call
x
Sale
$ 525
No Sale - $ 75
P(x)
.20
.80
x•P(x)
$ 105
- $ 60
E(x) = μ = ∑ xP(x) = $ 45
Econ10/Mgt 10
Stuffler
25
Binomial Distribution
Binomial distribution is the probability distribution
that results from doing a “binomial experiment”
which have the properties:
1.
2.
3.
4.
Fixed number of identical trials, represented as n.
Each trial has two possible outcomes:
a “success” or “failure”
For all trials, the probability of success, P(success)=p, and
the probability of failure, P(failure)=1–p=q, are constant.
The trials are independent
Econ10/Mgt 10
Stuffler
26
Several Binomial Distributions
Econ10/Mgt 10
Stuffler
27
Binomial Distribution
Success and failure: labels for binomial experiment
outcomes, no value judgment is implied.
EX: Coin flip results in either heads or tails.
If we define “heads” as success, then
“tails” is considered a failure.
Other binomial examples:
An election candidate wins or loses
An employee is male or female
A worker is employed or unemployed
Econ10/Mgt 10
Stuffler
28
Binomial Distribution
Binomial Formula:
p(x) =
n!
p x q n -x
x!(n - x)!
where x = number of successes in n trials,
n – x = number of failures in n trials,
px = the probability of success raised to the
number of successes, and
qn-x = probability of failure raised to the
number of failures
Econ10/Mgt 10
Stuffler
29
Binomial Distribution
The random variable of a binomial experiment is
defined as the number of successes in the n trials,
and is called the binomial random variable.
EX: Flip a fair coin 10 times
–1) Fixed number of trials  n=10
–2) Each trial has two possible outcomes  {heads (success),
tails (failure)}
–3) P(success)= 0.50; P(failure)=1–0.50 = 0.50 
–4) The trials are independent  (i.e. the outcome of heads on the
first flip will have no impact on subsequent coin flips).
Flipping a coin ten times is a binomial
experiment since all conditions are met.
Econ10/Mgt 10
Stuffler
30
Binomial Distribution
Another sales call example:
Assume a mutual fund salesman knows that there is
20% chance of closing a sale on each call he makes.
We want to determine the probability of making two sales in
three calls:
P(sale) = .2
P(no sale) = .8
P(X=2) =
3!
2!(3-2)!
.22.8.8 = .096
Econ10/Mgt 10
Stuffler
31
Binomial Distribution
Another sales call example:
Assume a mutual fund salesman knows that there is
20% chance of closing a sale on each call he makes
We want probability of making two sales in three calls
P(sale) = .2
P(no sale) = .8
GOOD NEWS!
Megastat→Probability→Discrete Distributions→Binomial
Econ10/Mgt 10
Stuffler
32
Binomial Distribution
Binomial distribution
3 n
0.2 p
X
0
1
2
3
P(X)
0.51200
0.38400
0.09600
0.00800
1.00000
cumulative
probability
0.51200
0.89600
0.99200
1.00000
0.600 expected value
0.480 variance
0.693 standard deviation
Econ10/Mgt 10
Stuffler
33
Pat Statsly
Pat Statsly is a student (not a good student) taking a
statistics course. Pat’s exam strategy is to rely on luck
for the first test. The test consists of 10 multiple-choice
questions. Each question has five possible answers, only
one of which is correct. Pat plans to guess the answer
to each question.
What is the probability that Pat gets no answers correct?
What is the probability that Pat gets two answers correct?
Econ10/Mgt 10
Stuffler
34
Pat Statsly
n=10
P(correct) = p = 1/5 = .20
P(wrong) = q = .80
Is this a binomial experiment?
Check the conditions
Econ10/Mgt 10
Stuffler
35
Pat Statsly
n=10
P(correct) = p = 1/5 = .20
P(wrong) = q = .80
Is this a binomial experiment? Check the conditions:
 There is a fixed finite number of trials (n=10).
 An answer can be either correct or incorrect.
 The probability of a correct answer (P(success)=.20) does
not change from question to question.
 Each answer is independent of the others.
Econ10/Mgt 10
Stuffler
36
Pat Statsly
n=10, and P(success) = .20
What is the probability that Pat gets no answers correct?
EX:
P(x=0)
What’s the interpretation of this result?
Econ10/Mgt 10
Stuffler
37
Pat Statsly
• n=10, and P(success) = .20
What is the probability that Pat gets no answers correct?
EX:
P(x=0)
Pat has about an 11% chance of getting no answers correct
using the guessing strategy.
Econ10/Mgt 10
Stuffler
38
Pat Statsly
Binomial distribution
10 n
0.2 p
X
0
1
2
3
4
5
6
7
8
9
10
P(X)
0.10737
0.26844
0.30199
0.20133
0.08808
0.02642
0.00551
0.00079
0.00007
0.00000
0.00000
1.00000
cumulative
probability
0.10737
0.37581
0.67780
0.87913
0.96721
0.99363
0.99914
0.99992
1.00000
1.00000
1.00000
2.000 expected value
1.600 variance
1.265 standard deviation
Econ10/Mgt 10
Stuffler
39
Pat Statsly
n=10, and P(success) = .20
What is the probability that Pat gets two answers correct?
EX: P(x=2)
Econ10/Mgt 10
Stuffler
40
Pat Statsly
We have been using the binomial probability distribution to
find probabilities for individual values of x.
To answer the question:
“Find the probability that Pat fails the quiz”
requires a cumulative probability, that is, P(X ≤ x)
If a grade on the quiz is less than 50% (i.e. 5 questions
out of 10), that’s considered a failed quiz.
Econ10/Mgt 10
Stuffler
41
Pat Statsly
We have been using the binomial probability distribution to
find probabilities for individual values of x.
To answer the question:
“Find the probability that Pat fails the test”
requires a cumulative probability, that is, P(X ≤ x)
If a grade on the test is less than 50% (i.e., 5 questions
out of 10), that’s considered a failed test.
We want to know what is: P(X ≤ 4) to answer
Econ10/Mgt 10
Stuffler
42
Pat Statsly
P(X ≤ 4) = P(0) + P(1) + P(2) + P(3) + P(4) = .9672
What is the interpretation of this result?
Econ10/Mgt 10
Stuffler
43
Pat Statsly
Its about 97% probable that Pat will
fail the test using the luck strategy
and guessing at answers.
Econ10/Mgt 10
Stuffler
44
Binomial Table
Calculating binomial probabilities by hand is tedious and
error prone. There is an easier way. Refer to Table E-6 in
the Appendices. For the Pat Statsly example, n=10, so go to
the n=10 table:
Look in the Column, p=.20 and substitute into:
P(X ≤ 4) = P(0) + P(1) + P(2) + P(3) + P(4)
P(X ≤ 4) = .1074 + .2684 + .3020 + .2013 + .0881 = .9672
The probability of Pat failing the test is 96.72%
Econ10/Mgt 10
Stuffler
45
Poisson Distribution
Named for Simeon Poisson, Poisson distribution -
Discrete probability distribution
There there are no trials
Number of independent events (successes) occurring in
a fixed time period or region of space that occur with a
known average rate such as, arrivals, departures, or
accidents, number of baskets in a quarter, etc.
Econ10/Mgt 10
Stuffler
46
Poisson Distribution
For example:
• The number of cars arriving at a service station in 1 hour.
(The interval of time is 1 hour.)
• The number of flaws in a bolt of cloth. (The specific region
is a bolt of cloth.)
• The number of accidents in 1 day on a particular stretch
of highway. (The interval is defined by both time, 1 day,
and space, the particular stretch of highway.)
Econ10/Mgt 10
Stuffler
47
Poisson Distribution
Poisson random variable - number of successes that
occur in a period of time or an interval of space
EX: On average, 96 trucks arrive at a border crossing
every hour.
EX: Number of typographical errors in a new textbook
edition averages 1.5 per 100 pages.
Econ10/Mgt 10
Stuffler
48
Poisson Distribution…
The Poisson random variable is the number of successes
that occur in a period of time or an interval of space in a
Poisson experiment.
successes
EX: On average, 96 trucks arrive at a border crossing
every hour.
time period
E.g. The number of typographic errors in a new textbook
edition averages 1.5 per 100 pages.
successes (?!)
interval
Econ10/Mgt 10
Stuffler
49
Poisson Distribution
Poisson experiment has four defining characteristics or
properties:
1. The number of successes that occur in any interval is
independent of the number of successes that occur in any
other interval
2. The probability of a success in an interval is the same for all
equal-size intervals
3. The probability of a success is proportional to the size of the
interval
4. Only one value is required to determine the probability of a
designated number of events occurring during an interval
Econ10/Mgt 10
Stuffler
50
Poisson Distribution
The probability that a Poisson random variable
assumes a value of x is given by:
µ = n(p) = number of successes times the probability of success
The unit of time or the interval should be short enough so that the
mean rate is not large (µ < 20)
YOUR TEXTBOOK USES λ (lambda) REPRESENT THE MEAN
NUMBER OF SUCCESSES IN THE INTERVAL
Econ10/Mgt 10
Stuffler
51
Poisson Distribution
EXAMPLE
The number of typographical errors in new editions of
textbooks varies considerably from book to book.
After some analysis she concludes that the number of
errors is Poisson distributed with a mean of 1.5 per
100 pages.
The instructor randomly selects 100 pages of a new
book. What is the probability that there are no typos?
MEGASTAT→PROBABILITY→DISCRETE DISTRIBUTIONS→POISSON
AND ENTER THE MEAN VALUE: 1.5, CLICK OK
Econ10/Mgt 10
Stuffler
52
Poisson Distribution
Poisson distribution
1.5 mean rate of occurrence
X
0
1
2
3
4
5
6
7
8
9
10
11
12
P(X)
0.22313
0.33470
0.25102
0.12551
0.04707
0.01412
0.00353
0.00076
0.00014
0.00002
0.00000
0.00000
0.00000
1.00000
cumulative
probability
0.22313
0.55783
0.80885
0.93436
0.98142
0.99554
0.99907
0.99983
0.99997
1.00000
1.00000
1.00000
1.00000
1.500 expected value
1.500 variance
1.225 standard deviation
Econ10/Mgt 10
Stuffler
53
Mean and Variance of a
Poisson Random Variable
If x is a Poisson random variable with parameter , then
Mean number of events per unit of time or space = µ
Variance = V(X) = σ2 = μ
Econ10/Mgt 10
Stuffler
54
Poisson Distribution
As mentioned on the first Poisson Distribution slide:
The probability of a success is proportional
to the size of the interval
Thus, knowing an error rate of 1.5 typos per 100
pages, we can determine a mean value for a 400
page book as:
μ = n(p) = 1.5(4) = 6 typos / 400 pages.
Econ10/Mgt 10
Stuffler
55
Poisson Distribution
For a 400 page book, what is the probability that there are
no typos?
P(X=0) =
Interpretation?
Econ10/Mgt 10
Stuffler
56
GOOD NEWS!
Go to Excel, Megastat, Probability, Discrete,
Poisson.
Type in the mean, μ, 6, then OK.
For a mean of 6,the Excel result is 0.00248
Econ10/Mgt 10
Stuffler
57
Poisson Distribution
For a 400 page book, what is the probability that
there are five or less typos?
P(X≤5) = P(0) + P(1) + … + P(5)
Another alternative is to refer to Table E-7 in the
Appendices
For x = 5, μ =6, and P(X ≤ 5) = P(0) + … + P(5) = .4456
There is about a 45% chance that there will be 5 or less typos
Econ10/Mgt 10
Stuffler
58
Hypergeometric Distribution
If a sample is taken from a finite
population without replacement, the
probability of X number of successes,
follows the hypergeometric distribution.
NOTE: When you see “without
replacement,” this indicates that
hypergeometric trials are dependent.
Econ10/Mgt 10
Stuffler
59
Hypergeometric Distribution
For both the Binomial Distribution
and the Hypergeometric
Distribution, there are two possible
outcomes for the random discrete variable:
Success or Failure
Econ10/Mgt 10
Stuffler
60
Hypergeometric Distribution
The difference is that for
Hypergeometric each trial is
“without replacement,” so the probability
changes from one draw to the next
For Binomial, the probability of success and
failure remain constant since each trial or
draw is independent
Econ10/Mgt 10
Stuffler
61
Hypergeometric Distribution
1.
2.
3.
4.
Trials are NOT independent
Finite population
Sample without replacement
Probability of success and failure
changes from trial to trial
Econ10/Mgt 10
Stuffler
62
Hypergeometric Distribution
r
P(x) = x
N–r
n–x
N
n
μ=nr
N
σ2 = r (N-r) n (N-n)
N2 (N-1)
where N = Total number in the population
n = number in the sample,
x = number selected or drawn,
r = number of successes in N
Econ10/Mgt 10
Stuffler
63
Hypergeometric Distribution
EXAMPLE: Assume that 3 stocks are
chosen randomly from a list of 10 stocks
and that of the 10 stocks 4 stocks pay
dividends. The number (x) of the three
selected stocks that pay a dividend is a
hypergeometric random variable.
Econ10/Mgt 10
Stuffler
64
Hypergeometric Distribution
N=10, n=3, r = 4 and
x = number of the 3 stocks selected
that pay a dividend
Calculate the mean, the variance and the
probability that none of the 3 stocks pay
dividends?
Econ10/Mgt 10
Stuffler
65
Hypergeometric Distribution
4 10 – 4
4!
6!
P(0) = 0
3-0 = 0!(4-0)! 3!(6-3)! = 1
10
10!
6
3
3!(10-3)!
μ = (3)(4) = 1.2
10
σ2 = 4(10-4)3(10-3) = .56, σ = .75
(10)2 (10-1)
Econ10/Mgt 10
Stuffler
66
Hypergeometric Distribution
GOOD NEWS!
Go to Excel→Megastat→Probability
→Discrete Probability Distributions
→Hypergeometric, enter the required
data
Econ10/Mgt 10
Stuffler
67
Hypergeometric Distribution
Hypergeometric distribution
10 N, population size
3 S, number of possible occurrences
4 n, sample size
X
0
1
2
3
P(X)
0.16667
0.50000
0.30000
0.03333
1.00000
cumulative
probability
0.16667
0.66667
0.96667
1.00000
1.200 expected value
0.560 variance
0.748 standard deviation
Econ10/Mgt 10
Stuffler
68
Discrete Probability
WHICH DISTRIBUTION?
Sony issued a recall for their plasma screen
TVs. They choose 25 TVs and 7 do not
operate. If the inspector chooses 3 at
random from the 25, what is the probability
that 2 of the 3 do not operate?
Econ10/Mgt 10
Stuffler
69
Discrete Probability
WHICH DISTRIBUTION?
When Wet Water drills a well, their
success rate is .55 and their profit is
$2,550. If they do not find water, they
lose $1,500.
Econ10/Mgt 10
Stuffler
70
Discrete Probability
WHICH DISTRIBUTION?
The US unemployment rate is
calculated from survey data of 60,000
households. What is the probability
that no more than 5,000 respondents
are unemployed?
Econ10/Mgt 10
Stuffler
71
Discrete Probability
WHICH DISTRIBUTION?
US Transportation and Safety Board
reports the annual average number of
plane crashes is 14.6. For a given
month, what is the probability that at 2
crashes will occur? That 3 crashes
will occur?
Econ10/Mgt 10
Stuffler
72
Discrete Probability
WHICH DISTRIBUTION?
The US Statistical Abstract of the US
tracks many data items such as, the
average number of automobiles per
household. The average for 2009 is
2.5 autos. What is the probability that
the 2010 average is at least 3 autos?
Econ10/Mgt 10
Stuffler
73
Discrete Random Variables
Summary:
Two Types of Random Variables
Discrete Probability Distributions
The Binomial Distribution
The Poisson Distribution
The Hypergeometric Distribution
Econ10/Mgt 10
Stuffler
74