SDA 3E Chapter 3 (1)
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Transcript of SDA 3E Chapter 3 (1)
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2007 Pearson Education
Chapter 3: Random Variablesand Probability Distributions
Part 1: Probability and Probability
Distributions
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Experiments and Outcomes Experiment the observation of some activity
or the act of taking a measurement
Roll dice Measure dimension of a manufactured good
Conduct market survey
Outcome a particular result of anexperiment
Sum of the dice
Length of the dimension
Proportion of respondents that favor a product
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Events Sample space - all possible outcomes of an
experiment Dice rolls of 2, 3, , 12 All real numbers corresponding to a dimension A proportion between 0 and 1
An event is a collection of one or moreoutcomes from Obtaining a 7 or 11 on a roll of dice Having a manufactured dimension larger or
smaller than the required tolerance The proportion of respondents that favor a
product is at least 0.60
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Probability Probability a measure (between 0 and 1) of
the likelihood that an event will occur
Three views: Classical definition: based on theory
Relative frequency: based on empirical data
Subjective: based on judgment
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Classical Definition Probability = number of favorable outcomes
divided by the total number of possible
outcomes Example: There are six ways of rolling a 7
with a pair of dice, and 36 possible rolls.Therefore, the probability of rolling a 7 is
6/36 = .167.
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Relative Frequency Definition Probability = number of times an event has
occurred in the past divided by the total
number of observations Example: Of the last 10 days when certain
weather conditions have been observed, ithas rained the next day 8 times. The
probability of rain the next day is 0.80
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Subjective Definition What is the probability that the New York
Yankees will win the World Series this year?
What is the probability your school will win itsconference championship this year?
What is the probability the NASDAQ will goup 3% next week?
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Probability Rules1. Probability associated with any outcome must be
between 0 and 1
2. Sum of probabilities over all possible outcomesmust be 1.0 Example: Flip a coin three times
Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH,TTT
Each has probability of (1/2)3 = 1/8
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Mutually Exclusive Events Two events are mutually exclusive if the
occurrence of any one means that none
of the other can occur at the same time.
CB
B & C
A
B
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Probability Calculations1. Probability of any event is the sum of
the outcomes that compose that event
2. If events A and B are mutually exclusive,then
P(A or B) = P(A) + P(B)
3. If events A and B are not mutuallyexclusive, thenP(A or B) = P(A) + P(B) P(A and B)
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Example What is the probability of obtaining exactly
two heads or exactly two tails in 3 flips of acoin? These events are mutually exclusive. Probability =
3/8 + 3/8 = 6/8
What is the probability of obtaining at leasttwo tails or at least one head? A = {TTT, TTH, THT, HTT}, B = {TTH, THT, THH,
HTT, HTH, HHT, HHH} The events are notmutually exclusive.
P(A) = 4/8; P(B) = 7/8; P(A&B) = 3/8. Therefore,
P(A or B) = 4/8 + 7/8 3/8 = 1
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Conditional Probability Conditional probability the probability
of the occurrence of one event, given
that the other event is known to haveoccurred.
P(A|B) = P(A and B)/P(B)
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Example What is the probability of exactly 2 heads in
three flips, given that the first flip is a head?
HHH, HHT, HTH, THH, HTT, THT, TTH, TTT A = exactly 2 heads in 3 flips
B = first flip is a head, P(B) = 4/8
A&B = {HHT, HTH} P(A&B) = 2/8
P(A|B) = P(A&B)/P(B) = (2/8)/(4/8) = i.e.:
B ={HHH, HHT, HTH, HTT};A shown in red
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Joint Probability Joint probability table probabilities of
outcomes of two different variables
Variable 1: Number of heads in three flipsof a coin
Variable 2: First flip (H or T)
Frequencies Joint Probabilities
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Marginal Probability
P(A|B) = P(A&B) / P(B) = (1/4)/(1/2) =
Marginalprobabilities
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Other Uses for Conditional
Probability P(A and B) = P(B|A)*P(A)
P(A) = P(A|B1)*P(B1) + P(A|B2)*P(B2) +
+ P(A|Bk)*P(Bk)
where B1, B2, , Bkare k mutuallyexclusive and exhaustive outcomes.
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Statistical Independence Two events A and B are statistically
independent if the probability of A given
B equals the probability of A; in otherwords P(A|B) = P(A) or P(B|A) = P(B).
Implications for marketing
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Bayes Theorem Suppose that A1, A2, , Ak is a set of
mutually exclusive and collectively
exhaustive events, and seek theprobability that some event Ai occursgiven that another event B has occurred.
P(B|Ai)*P(Ai)
P(Ai|B) = --------------------------------------------------------------------
P(B|A1)*P(A1) + P(B|A2)*P(A2) + + P(B|Ak)*P(Ak)
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Example Suppose that a digital camera manufacturer is
developing a new model. Historically, 70 percent ofcameras that have been introduced have been
successful it terms of meeting a required return oninvestment, while 30 percent have been unsuccessful.Analysis of past market research studies, conductedprior to worldwide market introduction, has found that90 percent of all successful product introductions had
received favorable consumer response, while 60percent of all unsuccessful products had also receivedfavorable consumer response. If the new modelreceives a favorable response from a market researchstudy, what is the probability that it will actually be
successful?
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Example (continued) Events
A1 = new camera successful, P(A1) = 0.7
A2 = new camera unsuccessful, P(A2) = 0.3 B1 = marketing response favorable, B2 = marketing response unfavorable
Other know information: P(B1|A1) = 0.9 P(B1|A2) = 0.6
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Applying Bayes TheoremP(B1|A1)*P(A1)
P(A1|B1) = -------------------------------------------------------------------------
P(B1
|A1
)*P(A1
) + P(B1
|A2
)*P(A2
) + + P(B1
|Ak
)*P(Ak
)
(0.9)(0.7)
P(A1|B1) = -------------------------------- = 0.778
(0.9)(0.7) + (0.6)(0.3)
Although 70 percent of all previous new models have beensuccessful, knowing that the marketing report is favorableincreases the likelihood to 77.8 percent.
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Random Variables Random variable a function that assigns a real
number to each element of a sample space (anumerical description of the outcome of an
experiment).Random variables are denoted by capitalletters, X, Y, ; specific values by lower case letters,x, y,
Random variables may be discrete, with a finite or
infinitely countable number of values (number ofinaccurate orders or number of imperfections on acar) or continuous, having any real value possiblywithin some limited range(tomorrows temperature)
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Examples of Random Variables Experiment: flip a coin 3 times.
Outcomes: TTT, TTH, THT, THH, HTT, HTH, HHT,
HHH Random variable: X = number of heads.
X can be either 0, 1, 2, or 3.
Experiment: observe end-of-week closing
stock price. Random variable: Y = closing stock price.
X can be any nonnegative real number.
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Probability Distributions Probability distribution a characterization of
the possible values a random variable may
assume along with the probability ofoccurrence.
Probability distributions may be defined forboth discrete and continuous random
variables.
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Discrete Random Variables Probability mass functionf(x): specifies the
probability of each discrete outcome
Two properties: 0 f(xi) 1f(xi) = 1
Cumulative distribution function, F(x): specifiesthe probability that the random variable will beless than or equal tox.
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Example: Census Data
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Charts for f(x) and F(x)
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Example: Roll Two DiceSumofdice, x
2 3 4 5 6 7 8 9 10 11 12
f(x) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
F(x) 1/36 3/36 6/36 10/36 15/36 21/36 26/36 30/36 33/36 35/36 1.0
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Joint Probability Distributions Joint probability distribution of two random variables,
f(x,y): specifies that probability that X = x and Y = ysimultaneously.
Not a High
School Grad
High School
Graduate
Some College
No Degree
Associate's
Degree
Bachelor's
Degree
Advanced
Degree
Age
25-34 0.027 0.073 0.045 0.020 0.049 0.014 0.228
35-44 0.031 0.088 0.047 0.024 0.047 0.021 0.258
45-54 0.026 0.063 0.035 0.017 0.035 0.022 0.198
55-64 0.026 0.048 0.019 0.007 0.017 0.012 0.129
65-74 0.030 0.038 0.014 0.004 0.010 0.007 0.104
75 and older 0.031 0.027 0.011 0.003 0.007 0.004 0.082
0.172 0.338 0.172 0.075 0.164 0.080 1.000
Joint Marginal
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Continuous Random Variables Probability density function, f(x), a continuous
function that describes the probability of
outcomes for the random variable X. Ahistogram of sample data approximates theshape of the underlying density function.
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Properties of Probability
Density Functions f(x) 0 for all x
Total area under f(x) = 1
There are always infinitely many values for X P(X = x) = 0
We can only define probabilities over
intervals: e.g.,P( c < X < d), P(X < c), or P(X > d)
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Cumulative Distribution
Function F(x) specifies the probability that the random
variable X will be less than or equal to x; that
is, P(X x). F(x) is equal to the area under f(x) to the left
of x
The probability that X is between a and b isthe area under f(x) from a to b = F(b) F(a)
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Example
f(x) = 4x 16 for 4 x 4.5
= -4x + 20 for 4.5 x 5
F(x) = 2x2 16x + 32 for 4 x 4.5
= -2x2 + 20x 49 for 4.5 x 5
P(X > 4.7) = F(5) F(4.7) = 1 - 0.82 = 0.18
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Cumulative Distribution Function
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Expected Value and Variance
of Random Variables Expected value of a random variable X is the theoretical
analogy of the mean, or weighted average of possiblevalues:
Variance and standard deviation of a random variable X:
1=i
ii )f(xx=E[X]
1j =
j2
j )f(xE[X])-(x=Var[X]
1j =
j
2
jX )f(xE[X])-(x=
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Example You play a lottery in which you buy a ticket
for $50 and are told you have a 1 in 1000
chance of winning $25,000. The randomvariable X is your net winnings, and itsprobability distribution is
x f(x)
-$50 0.999
$24,950 0.001
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Calculations E[X] = -$50(0.999) + $24,950(0.001) =
-$25.00
Var[X] = (-50 - [-25.00])2(0.999) +(24,950 - [-25.00])2(0.001) = 624,375
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Functions of Random
Variables Two useful formulas:
E[aX] = aE[X]
Var[aX] = a2Var[X]
where a is any constant
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Discrete Probability
Distributions Bernoulli
Binomial
Poisson
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Bernoulli Distribution A random variable with two possible
outcomes (x = 0 and x = 1), each with
constant probabilities of occurrence:
f(x) = p if x = 1
f(x) = 1 p if x = 0
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Binomial Distribution n independent replications of a Bernoulli trial
with constant probability of success p on eachtrial
Expected value = np
Variance = np(1-p)
x
n
= )!(!!
xnx
n
otherwise
nxforppxnxf xnx
0
,...,2,1,0)1()(
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Example If the probability that any individual will react
positively to a new drug is 0.8, what is the probabilitydistribution that 4 individuals will react positively out
of a sample of 10
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Excel Function BINOMDIST(number_s, trials, probability_s,
cumulative)
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Examples of the Binomial
Distribution
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Poisson Distribution Models the number of occurrences in some unit of
measure, e.g., events per unit time, number of itemsper order
X = number of events that occur; x = 0, 1, 2,
Expected value = l; variance = l
Poisson approximates binomial when n is large and psmall
otherwise
xforx
exf
x
0
,...2,1,0!
)(
ll
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Example Suppose that the average number of customers
arriving at an ATM during lunch hour is l = 12customers per hour. The probability that exactly x
customers will arrive during the hour is
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Excel Function POISSON (x, mean, cumulative)
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Poisson Distribution (l = 12)
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Properties of Continuous
Distributions Continuous distributions have one or
more parameters that characterize the
density function: Shape parameter controls the shape of
the distribution
Scale parameter controls the unit of
measurement Location parameter specifies the location
relative to zero on the horizontal axis
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Uniform Distribution Density function
Distribution function
f xb a
if a x b( )
1
0
1
if x a
F xx a
b aif a x b
if b x
( )
a b
x
f(x)m = (a + b)/22 = (b a)2/12
a = location
b
a = scale
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Normal Distribution Familiar bell-shaped curve.
Symmetric, median = mean = mode; half the area is oneither side of the mean
Range is unbounded: the curve never touches the x-axisParameters
Mean, m (location)
Variance 2 > 0 (scale)
Density function:
f(x) =e
-(x - )2
m
2
2
2
2
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Standard Normal Distribution Standard normal: mean = 0, variance = 1,
denoted as N(0,1)
See Appendix Table A.1
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Standardized Normal Values Transformation from N(m,) to N(0,1):
Standardized z-values are expressed in units
of standard deviations of X.
mx=z
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Areas Under the Normal
Density About 68.3% is within one sigma of the mean
About 95.4% is within two sigma of the mean
About 99.7% is within three sigma of themean
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Normal Probability
Calculations Customer demand averages 750
units/month with a standard deviation of
100 units/month. Find P(X>900), P(X>700), P(700
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P(X>900)5.1
100
750900=z
Area = 0.9332
Area = 0.0668
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P(X>700)
5.0100
750700=z
P(X > 700) = P(Z > -0.5) = 1 - P(Z < -0.5).From Table A.1, P(Z < -0.5) = 0.3085.
Thus, P(X > 700) = 1 - 0.3085 = 0.6915.
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P(700
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P(X>x)>.10
28.1100
750=z
xFirst, find the valueof z from Table A.1,then solve for x
x = 878
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Excel Support
NORMDIST(x, mean, standard_deviation,cumulative)
NORMSDIST(z): same as Table A.1 STANDARDIZE(x, mean, standard_deviation)
computes z-values
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Computing Normal Probabilities
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Triangular Distribution
Three parameters:
Minimum, a
Maximum, b
Most likely, c
a is the location parameter;
(b a) the scale parameter,
c the shape parameter.
2
2
0
( )
( )( )
( )( )
( )( )
x a
b a c aif a x c
f xb x
b a b cif c x b
otherwise
Mean = (a + b + c)/3
Variance = (a2
+ b2
+ c2
ab
ac
bc)/18
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Exponential Distribution
Models events that occur randomly over time
Customer arrivals, machine failures
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Properties of Exponential
Density function
Distribution function
Mean = 1/l Variance = 1/l2
Excel function EXPONDIST(x, lambda, cumulative).
f(x) = le-lx, x 0
F(x) = 1 - e-lx, x 0
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Lognormal Distribution
X is lognormal if the distribution of lnX isnormal
Positively skewed, bounded below by 0 Models task times, stock and real estate
prices
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Gamma Distribution
Family of distributions defined by shape a,scale b, and location L
Defined for x > L Models task times, time between events,
inventories
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Weibull Distribution
Family of distributions defined by shape a,scale b, and location L
When L = 0 and b = 1, same as exponentialwith l = 1/a.
Models results from life tests, equipmentfailures, and task times.
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Beta Distribution
Defined over the range (0, s)
Two shape parameters a and b; if equal, beta
is symmetric; a < b, positively skewed; ifeither equals 1 and the other > 1, Jshaped.
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Geometric Distribution
Sequence of Bernoulli trials that describes thenumber of trials until the first success.
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Negative Binomial
Distribution of number of trials until the rthsuccess.
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Logistic Distribution
Describes the growth of a population overtime.
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Pareto Distribution
Distribution in which a small proportion ofitems accounts for a large proportion of somecharacteristic.
PHSt t T l P b bilit &
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PHStatTool: Probability &Probability Distributions
PHStatmenu > Probability & ProbabilityDistributions> choice of distribution:
Normal
Binomial
Exponential
Poisson
Hypergeometric
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PHStatOutput