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Chapter 11: Selected
Quantitative
Relationships (pt. 1)
ISE 443 / ETM 543
Fall 2013
The development of large, complex systems
involves the use of quantitative relationships
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To predict system performance before it is built
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response times
success rates
system availability
system reliability
error rates
catastrophic failure
Because of the uncertainty involved, probability is the
most used quantitative relationship in systems
engineering (and, to a lesser extent, project
management)
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To start, we’ll review the fundamentals
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Basic probability – Ned and Ryan S.
Discrete distributions
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Continuous distributions
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binomial distribution – Lexie and Rebecca
poisson distribution – Filipe, Xin, and Yuan
normal distribution – Alfred and James
uniform distribution – Justin and Gleidson
exponential distribution – Elvire and Ryan K.
Means and variances – Jamie and Geneve
Sums of variables – Isis and Jessica
Functions of random variables – Charmaine and Melanie
Two variables (joint probability) – Celso and David
Correlation – Lucas and Kyle
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Your turn …

Form into pairs. As a pair review the list on the previous
slide and select a topic you wish to review for the class.
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Once you have decided on a topic, STAND UP. When
you are recognized (I will go in order) state your selected
topic.
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You may want to have at least one “backup” topic.
If someone else selects your topic, move to your “backup”. If you
don’t have a backup, sit down until you have selected another.
Review the section in the book related to your topic and
develop a brief summary to share with the class today
(you will have 10 minutes to prepare).
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10 minutes after the last topic is selected, we will go through each
topic as a class and discuss what you have so far.
Make note of the feedback you receive. You may find this useful
when completing the homework assignment.
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Basic probability
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For any given event P > 0
For a certain event, P = 1, if it can’t happen P=0,
if it may happen it’s between 0 and 1.
If 2 events are mutually exclusive, add the
probabilities to find the probability of either one
happening
To find probability of BOTH happening, multiply
the 2 probabilities
Other rules for independence, non-independent,
etc.
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Basic Probability
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For any event, the possibility of it occurring is
greater than or equal to zero. If it’s certain to
happen, the probability of it occurring is 1. If it’s
completely impossible, the probability of it occurring
is 0. Anything else has a probability of between 1
and 0. For example, if you roll a dice there is a 1/6
probability of getting any particular number
To find the probability of at least one of two
mutually exclusive events happening, add the odds
together. For example, if you roll a dice there is a
1/6 chance to get a 5 and a 1/6 chance to get a 4.
Therefore, there is a 2/6 (1/3) chance of getting
either a 5 or a 4.
By Ryan Stapleton and Ned
Nobles
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If you want to get both of two possibilities,
multiply the odds together. For example, if you
roll 2 dice, the odds of getting a 5 and a 4 are
1/6 *1/6, or 1/36.
By Ryan Stapleton and Ned
Nobles
Discrete distributions: Binomial
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Arises when there are repeated independent
trials with only 2 possible results
If P(success) = p and P(failure) = q, then p+q=1
The distribution is the equation 11.23 on page
345.
The distribution defines the probability of exactly
x successes in n independent trials
Example at top of page 346
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Binomial Distribution
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Arises with repeated independent trials with only
two possible outcomes
P(success)=p P(failure)=q p+q=1
𝑛 𝑥 𝑛−𝑥
𝑃 𝑥 =
𝑝 𝑞
, for x= 0,1,2,3,…n
𝑥
= 0, otherwise
The distribution defines the probability of exactly
x successes in n independent trials
Lexy Blaha & Rebecca King
Example
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If, when throwing a die, an odd number is
success and an even number failure, the
probability of exactly 4 success in 10 trials is
10 4 (10−4)
P(4)=
.5 .5
=.205
4
Lexy Blaha & Rebecca King
Real World Example
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In a multiple-choice exam with 4 choices for each question, what is
the probability that a student gets exactly 2 correct if he chooses the
answers randomly. Assume there are 10 questions in all.
n= 10
p= 1/4 = 0.25
We want P(x = 2 ) , where x is the number of correct answers out of
10.
10
𝑃 𝑥 =
.252 .75(10−2)
2
= 0.2816
Mean = np = 10(1/4) = 2.5
Standard deviation = 𝑛𝑝(1 − 𝑝) = 1.3693
Lexy Blaha & Rebecca King
Discrete distributions: Poisson
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Deals with the issue of software reliability
May be used in situations for which events
happen at some rate and we wish to ascertain
the probability
Example on page 346
Formula is also on 346
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The Poisson Distribution
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The Poisson is a discrete distribution given by
the following formula:
(𝑡)𝑘 exp(−𝑡)
P(k) =
for 𝑘 = 0, 1, 2, … , n
𝑘!
Where P(k) is the probability of exactly k events
of interest,  is the rate at which such events
are occurring, and t is the time over which the
events are occurring.
Team: Xin, Yuan, Filipe
The Poisson Distribution
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The distribution may be used in situations for
which events happen at some rate and we wish
to ascertain the probability of some number of
events occurring in a total period of time, or in a
certain space.
Team: Xin, Yuan, & Filipe
Example
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Cars are passing a toll booth at an overall rate of
about 120 cars per hour. The probability that
exactly three cars will pass through the toll booth
in a period of 1 minute would be
P(3) =
[ 2 1 ]3 exp[− 2 1 ]
=
3!
0.18
Team: Xin, Yuan, & Filipe
Continuous distributions: Normal
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AKA Gaussian
Bell-shaped curve
Density function on page 347, eq. 11.25
Integrate (p(x < X))
OR use tables (p(z <Z))
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Normal Distribution
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Gaussian Distribution
To calculate
probabilities, one must
integrate the density
function
This calculates the
probability that x is less
than or equal to a
specified number
However, normal
distribution is not readily
integrable so we must
resort to using a Z-table.
Albert Sykes & James
Edwards
Z-Table
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Albert Sykes & James
Edwards
Example
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Albert Sykes & James
Edwards
Continuous distributions: Uniform
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Occurs when there is a constant probability over
a range
Cumulative density function is a triangle from 0
to 1.
Equations for the mean and variance are on
page 350
Example on page 350 (arrow shot at bullseye)
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Uniform Distribution
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The Uniform Distribution is useful in random
number generation.
Random Numbers are very useful in: gambling
statistical sampling, cryptography, and anything
seeking an unpredictable result.
Source: http://www.giackop.com/blog/wpcontent/uploads/numbers.jpg
Justin Blount - Gleidson
Gurgel - ISE 443 Project
Management
Uniform Distribution
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Its graph is ‘flat’ over its entire range.
Mean: (a+b)/2
Variance: (b-a)^2/12
PDF:
http://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Uniform
_Distribution_PDF_SVG.svg/350pxUniform_Distribution_PDF_SVG.svg.png
CDF:
upload.wikimedia.org/wikipedia/commons/thumb/6/63/Uniform_cdf.svg/350pxUniform_cdf.svg.png
Justin Blount - Gleidson
Gurgel - ISE 443 Project
Management
Uniform Distribution
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Applications:
 Statistics:
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Randomness is commonly used to create simple random
samples. This lets surveys of completely random groups of
people provide realistic data.
 Computer
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Simulation:
Simulation of random events require random numbers.
 Gambling:
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Gambling theory is based on random numbers.
Justin Blount - Gleidson
Gurgel - ISE 443 Project
Management
Continuous distributions: Exponential
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Density and cumulative dist function are
illustrated on pg 350
The CDF starts at 0 and approaches 1
asymptotically
Formulas are given on pg 350
Widely used in reliability theory wherein the
value of lambda is taken to be a constant failure
rate
Failure rate and MTBF are reciprocals of each
other
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The Exponential Distribution
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a.k.a radical exponential distribution, it is the probability
distribution that describes the time between events in a
Poisson process
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The probability density function (pdf) of an exponential
distribution is:
•
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p(x)= λe-λx , for x ≥ 0
=0
, for x < 0
The cumulative distribution function (cdf) starts at zero
and approaches the value of unity asymptotically
Elvire Koffi
Ryan King
The Exponential Distribution
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The cumulative distribution function is given by:
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F(x) = 1 – e-λx , for x ≥ 0
=0
, for x < 0
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The exponential distribution is widely used in reliability
theory wherein the value of λ is taken to be a constant
failure rate for a part of a system
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The failure rate and the mean time between failures
(MTBF or 1/λ) are reciprocals of one another
Elvire Koffi
Ryan King
The Exponential Distribution
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Example
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Suppose that the amount of time one spends in a bank is
exponentially distributed with mean 10 minutes, λ = 1/10.
What is the probability that a customer will spend more
than 15 minutes in the bank?
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Solution:
P(x > 15) = e-15λ = e-3/2 = 0.22
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Elvire Koffi
Ryan King
Means and variances
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Mean equation is on pg 340 (both discrete and
continuous)
Example of rolling dice
Variance eq on pg 341
Variance indicates the spread of the distribution
Std deviation is the square root of the variance
A critical performance measure is the S/N ratio,
which is …
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Mean Value (pg.340)
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Discrete: X = m(X) = ΣX P(X) for all X
Continuous: x = m(x) = x p(x)dx
EXAMPLE:
• The mean value of rolling a single die is:
X=1(1/6)+2(1/6)+3(1/6)+4(1/6)+5(1/6)
+6(1/6)
X=3.5
• Note-This result leads to a mean value that is
different from all possible values of the variable.
Duffy & Lopez
Variance and Standard Deviation (pg.341)
VARIANCE
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Discrete: σ2 =Σ(X-m)2 P(X) for all X
Continuous: σ2 =  (x-m)2 p(x)dx
EXAMPLE:
• The variance of rolling a single die is:
σ2=(1-3.5)2(1/6)+(23.5)2(1/6)+…+(6-3.5)2(1/6) = 2.9
STANDARD DEVIATION
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The square root of the variance √σ2 =σ
EXAMPLE:
• Also known as the root-mean-squared:
√σ2= √2.9 = 1.7029
Duffy & Lopez
Signal-to-Noise Ratio (pg. 341)
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This is a ratio of the signal power over the noise power: S/N
Equivalent to the square of the signal value to the variance
of the noise distribution.
EXAMPLE: This statistic can be related to classify accuracy
given in an ideal linear discrimination, such as the human
eyes ability to detect color signals.
http://www.dspguide.com/ch25/3.htm
Duffy & Lopez
Sums of variables
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Probability of the sum = sum of the probabilities
(when the variables are mutually exclusive)
Eq on pg 341
Distribution of the sum is the convolution of the
individual distributions
Variance of the sum is the sum of the variances
only when the individual distributions are
independent
The mean value of the sum is the sum of the
mean values
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Sums of variables
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If A and B are mutually exclusive, then:
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The distribution of a sum is the convolution of the individual
distributions
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The sum of 2 uniform distributions lead to a triangular distribution
The mean value of a sum is the sum of the mean values:
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P(A or B) = P(A +B) = P(A) + P(B)
Mean(Z) = E(Z) = mean(X + Y) = mean(X) + mean(Y)
The variance of a sum is the sum of the variances, only if the
individual distributions are independent. That is, only when
P(XY) = G(X)H(Y).
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σ²(Z) = σ²(X + Y) = σ²(X) + σ²(Y) when X and Y are independent
Isis Corrêa
Jéssica Bento
1
Functions of random variables
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Arise when we make a functional
transformationa dn wish to examing the result
Value is to track how a varaible behaves as it is
processed through a system
Example on pg 343
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Functions of Random Variables
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One of the values of understanding the transformation of random
variables (x) is to track how such a variable behaves as it is
processed through a system (y) (pg. 344)
In terms of y = mx+b, it is important to look at x because the input,
in conjunction with the constants (m & b), determines the output (y)
and its behavior
From a Systems Engineering and Project Management standpoint, it
is important to realize how the input variable (x), reacts with the
constraints of the system to produce an output (y).
Markman and Robinson
Functions of Random Variables Example
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Consider a random roll of two dice. There we defined the random variable X to represent the
sum of the values on the two rolls. Now let h(x) = |x-7|, so that h(x) ≡ |X-7| represents the
absolute difference between the observed sum of the two rolls and the average value 7. Then
h(X) has a pmf on a new probability space S2 ≡ {0, 1, 2, 3, 4, 5}. In this case we get the pmf
of h(X) being ph(x) (k) ≡ P(h(X) = k) ≡ P({s S: h(X(s)) = k}) for k S2 where,
ph(x)(5) = P(h(X) = 5) ≡ P(|X-7| = 5) = 2/36 = 1/18
ph(x)(4) = P(h(X) = 4) ≡ P(|X-7| = 4) = 4/36 = 2/18
ph(x)(3)= P(h(X) = 3) ≡ P(|X-7| = 3) = 6/36 = 3/18
ph(x)(2) = P(h(X) = 2) ≡ P(|X-7| = 2) = 8/36 = 4/18
ph(x)(1) = P(h(X) = 1) ≡ P(|X-7| = 1) = 10/36 = 5/18
ph(x)(0) = P(h(X) = 0) ≡ P(|X-7| = 0) = 6/36 = 3/18
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In this setting we can compute probabilities for events associated with h(X) ≡ |X-7| in three
ways: using each of the pmf’s p, px, and ph(x).
Markman and Robinson
Two variables (joint probability)
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Explains the joint behavior of 2 independent
varialbes where the prob of both happening can
be exrpessed by P(x,y)= P(x)P(y)
Example on pg 344
Mean or expected value eq on the same page
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Joint probability of two random variables
If p(x,y)=g(x)h(y) then the two variables are
independent
You can calculate the chances of two or more
independent events happening by multiplying
the probability of the events.
Probability of A and B equals the probability of
A times the probability of B
Celso Pereira David
Rodriguez
Example two variable joint probability
Example: Probability of 2 Heads using two coins
For each toss of a coin a "Head" has a probability
of 0.5
And so the chance of getting 2 Heads is 0.25
Celso Pereira David
Rodriguez
Expected value and Variance
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The expected value and, consequently, the
variance of a function of random variables can
be obtained using the following formulas
Expected Value
Variance
Celso Pereira David
Rodriguez
Correlation
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2 variables have a relationship
Correlation coefficient eq on pg 345 (11.22)
Normalized to values between -1 and +1
+1 perfect positive correlation, 0 is no
correlation, -1 is perfect neg correlation
Found
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Correlation
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If two variables x and y have a relationship, they
correlate to one another;
The correlation coefficient is given by:
Lucas Meyer & Kyle Adair
Correlation
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The correlation coefficient is normalized to
values between -1 and +1, where:
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+1 = perfect positive correlation;
0 = no correlation;
-1 = perfect negative correlation.
Lucas Meyer & Kyle Adair
Correlation Example
Data
Solution
Lucas Meyer & Kyle Adair
Source: http://www.mathsisfun.com/data/correlation.html