Transcript statistics for business and economics

Discrete Probability Distributions
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Random Variables
Discrete Probability Distributions
Expected Value and Variance
Binomial Probability Distribution
Poisson Probability Distribution
.40
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.20
.10
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Random Variables
A random variable is a numerical description of the
outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
Example: JSL Appliances
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Discrete random variable with a finite number
of values
Let x = number of TVs sold at the store in one day,
where x can take on 5 values (0, 1, 2, 3, 4)
Example: JSL Appliances
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Discrete random variable with an infinite sequence
of values
Let x = number of customers arriving in one day,
where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no
finite upper limit on the number that might arrive.
Random Variables
Question
Family
size
Random Variable x
x = Number of dependents in
family reported on tax return
Type
Discrete
Distance from x = Distance in miles from
home to store home to the store site
Continuous
Own dog
or cat
Discrete
x = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
Discrete Probability Distributions
The probability distribution for a random variable
describes how probabilities are distributed over
the values of the random variable.
We can describe a discrete probability distribution
with a table, graph, or equation.
Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by f(x), which provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
f(x) > 0
f(x) = 1
Example: JSL Appliances
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Using past data on TV sales, …
a tabular representation of the probability
distribution for TV sales was developed.
Units Sold
0
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2
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4
Number
of Days
80
50
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10
20
200
x
0
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2
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4
f(x)
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.25
.20
.05
.10
1.00
80/200
Example: JSL Appliances
Graphical Representation of the Probability
Distribution
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Probability
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.20
.10
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Values of Random Variable x (TV sales)
Discrete Uniform Probability Distribution
The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
f(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume
Expected Value and Variance
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) =  = xf(x)
The variance summarizes the variability in the
values of a random variable.
Var(x) =  2 = (x - )2f(x)
The standard deviation, , is defined as the positive
square root of the variance.
Example: JSL Appliances
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Expected Value of a Discrete Random Variable
x
0
1
2
3
4
f(x)
xf(x)
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.00
.25
.25
.20
.40
.05
.15
.10
.40
E(x) = 1.20
expected number of
TVs sold in a day
Example: JSL Appliances
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Variance and Standard Deviation
of a Discrete Random Variable
x
x-
0
1
2
3
4
-1.2
-0.2
0.8
1.8
2.8
(x - )2
f(x)
(x - )2f(x)
1.44
0.04
0.64
3.24
7.84
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.20
.05
.10
.576
.010
.128
.162
.784
Variance of daily sales =  2 = 1.660
TVs
squared
Standard deviation of daily sales = 1.2884 TVs
Using Excel to Compute the Expected
Value, Variance, and Standard Deviation
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Formula Worksheet
A
Sales
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B
Probability
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0.20
0.05
0.10
Mean =SUMPRODUCT(A2:A6,B2:B6)
Variance =SUMPRODUCT(C2:C6,B2:B6)
Std.Dev. =SQRT(B9)
C
Sq.Dev.from Mean
=(A2-$B$8)^2
=(A3-$B$8)^2
=(A4-$B$8)^2
=(A5-$B$8)^2
=(A6-$B$8)^2
Using Excel to Compute the Expected
Value, Variance, and Standard Deviation
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Value Worksheet
A
Sales
0
1
2
3
4
Mean 1.2
Variance 1.66
Std.Dev. 1.2884
B
Probability
0.40
0.25
0.20
0.05
0.10
C
Sq.Dev.from Mean
1.44
0.04
0.64
3.24
7.84
Binomial Probability Distribution
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Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
stationarity
assumption
4. The trials are independent.
Binomial Probability Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
Binomial Probability Distribution
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Binomial Probability Function
n!
f (x) 
p x (1  p)( n  x )
x !(n  x )!
where:
f(x) = the probability of x successes in n trials
n = the number of trials
p = the probability of success on any one trial
Binomial Probability Distribution
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Binomial Probability Function
n!
f (x) 
p x (1  p)( n  x )
x !(n  x )!
n!
x !(n  x )!
Number of experimental
outcomes providing exactly
x successes in n trials
(nx )
p (1  p)
x
Probability of a particular
sequence of trial outcomes
with x successes in n trials
Example: Evans Electronics
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Binomial Probability Distribution
Evans is concerned about a low retention rate for
employees. In recent years, management has seen a
turnover of 10% of the hourly employees annually.
Thus, for any hourly employee chosen at random,
management estimates a probability of 0.1 that the
person will not be with the company next year.
Example: Evans Electronics
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Binomial Probability Distribution
Choosing 3 hourly employees at random, what is
the probability that 1 of them will leave the company
this year?
Let: p = .10, n = 3, x = 1
n!
f ( x) 
p x (1  p ) (n  x )
x !( n  x )!
3!
f (1) 
(0.1)1 (0.9)2  3(.1)(.81)  .243
1!(3  1)!
Example: Evans Electronics
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Tree Diagram
1st Worker
2nd Worker
Leaves (.1)
Leaves
(.1)
3rd Worker
L (.1)
x
3
Prob.
.0010
S (.9)
2
.0090
L (.1)
2
.0090
S (.9)
1
.0810
L (.1)
2
.0090
S (.9)
1
.0810
1
.0810
0
.7290
Stays (.9)
Leaves (.1)
Stays
(.9)
L (.1)
Stays (.9)
S (.9)
Using Excel to Compute
Binomial Probabilities
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Formula Worksheet
A
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x
0
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B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
f (x )
=BINOMDIST(A5,$A$1,$A$2,FALSE)
=BINOMDIST(A6,$A$1,$A$2,FALSE)
=BINOMDIST(A7,$A$1,$A$2,FALSE)
=BINOMDIST(A8,$A$1,$A$2,FALSE)
Using Excel to Compute
Binomial Probabilities
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Value Worksheet
A
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x
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B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
f (x )
0.729
0.243
0.027
0.001
Using Excel to Compute
Cumulative Binomial Probabilities
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Formula Worksheet
A
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x
0
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B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
Cumulative Probability
=BINOMDIST(A5,$A$1,$A$2,TRUE)
=BINOMDIST(A6,$A$1,$A$2,TRUE)
=BINOMDIST(A7,$A$1,$A$2,TRUE)
=BINOMDIST(A8,$A$1,$A$2,TRUE)
Using Excel to Compute
Cumulative Binomial Probabilities
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Value Worksheet
A
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2
3
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5
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7
8
9
x
0
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3
B
3 = Number of Trials (n )
0.1 = Probability of Success (p )
Cumulative Probability
0.729
0.972
0.999
1.000
Binomial Probability Distribution
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Expected Value
E(x) =  = np
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Variance
Var(x) =  2 = np(1  p)
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Standard Deviation
  np(1  p)
Binomial Probability Distribution
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Binomial Probability Distribution
• Expected Value
E(x) =  = 3(.1) = .3 employees out of 3
• Variance
Var(x) =  2 = 3(.1)(.9) = .27
• Standard Deviation
  3(.1)(.9)  .52 employees
Poisson Probability Distribution
A Poisson distributed random variable is often
useful in estimating the number of occurrences
over a specified interval of time or space
It is a discrete random variable that may assume
an infinite sequence of values (x = 0, 1, 2, . . . ).
Poisson Probability Distribution
Examples of a Poisson distributed random variable:
the number of knotholes in 24 linear feet of
pine board
the number of vehicles arriving at a
toll booth in one hour
Poisson Probability Distribution
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Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence in any
interval is independent of the occurrence or
nonoccurrence in any other interval.
Poisson Probability Distribution
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Poisson Probability Function
f ( x) 
 x e 
x!
where:
f(x) = probability of x occurrences in an interval
 = mean number of occurrences in an interval
e = 2.71828
Example: Mercy Hospital
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Poisson Probability Function
MERCY
Patients arrive at the
emergency room of Mercy
Hospital at the average
rate of 6 per hour on
weekend evenings.
What is the
probability of 4 arrivals in
30 minutes on a weekend evening?
Example: Mercy Hospital
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Poisson Probability Function
 = 6/hour = 3/half-hour, x = 4
34 (2.71828)3
f (4) 
 .1680
4!
MERCY
Using Excel to Compute
Poisson Probabilities
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MERCY
Formula Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
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1
6
2
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3
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4
9
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10
6
… and so on
Probability f (x )
=POISSON(A4,$A$1,FALSE)
=POISSON(A5,$A$1,FALSE)
=POISSON(A6,$A$1,FALSE)
=POISSON(A7,$A$1,FALSE)
=POISSON(A8,$A$1,FALSE)
=POISSON(A9,$A$1,FALSE)
=POISSON(A10,$A$1,FALSE)
… and so on
Using Excel to Compute
Poisson Probabilities
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MERCY
Value Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
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3
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4
9
5
10
6
… and so on
Probability f (x )
0.0498
0.1494
0.2240
0.2240
0.1680
0.1008
0.0504
… and so on
Example: Mercy Hospital
Poisson Distribution of Arrivals
Poisson Probabilities
0.25
Probability

MERCY
0.20
actually,
the sequence
continues:
11, 12, …
0.15
0.10
0.05
0.00
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7
8
9
Number of Arrivals in 30 Minutes
10
Using Excel to Compute
Cumulative Poisson Probabilities
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MERCY
Formula Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
7
3
8
4
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5
10
6
… and so on
Cumulative Probability
=POISSON(A4,$A$1,TRUE)
=POISSON(A5,$A$1,TRUE)
=POISSON(A6,$A$1,TRUE)
=POISSON(A7,$A$1,TRUE)
=POISSON(A8,$A$1,TRUE)
=POISSON(A9,$A$1,TRUE)
=POISSON(A10,$A$1,TRUE)
… and so on
Using Excel to Compute
Cumulative Poisson Probabilities

MERCY
Value Worksheet
A
1
2
B
3 = Mean No. of Occurrences ( )
Number of
3 Arrivals (x )
4
0
5
1
6
2
7
3
8
4
9
5
10
6
… and so on
Cumulative Probability
0.0498
0.1991
0.4232
0.6472
0.8153
0.9161
0.9665
… and so on