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Transcript of Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
CHAPTER 3:CONTINUOUS DISTRIBUTIONS
AND PROBABILITY PLOTS
Instructor:Wandaliz Torres-García, Ph. D.
Reference: Most slides adapted from Montgomery et al. (2011)
ININ 5559ENGINEERING STATISTICS
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
RANDOM VARIABLES AND PROBABIDISTRIBUTIONS1. Random Numbers, Random Variables (RVs), Random Experiment (3-1 & 3-2)
2. Probability Laws (3-3)
3. Continuous RVs (3-4 & 3-5)CRVs Distributions: Normal, Lognormal, Gamma, Weibull, Beta, and others
4. Probability Plots (3-6)
5. Discrete RVs (3-7 & 3-8)DRVs Distributions: Binomial
6. Poisson Process (3-9)
7. Normal Approximation to the Binomial and Poisson Distributions (3-10)
8. More than one RV and Independence (3-11)
9. Functions of RVs (3-12)
10. Statistics and the Central Limit Theorem (3-13)
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
Undoubtedly, the most widely used model for the distributionof a random variable is anormal distribution .
• Central limit theorem• Gaussian distribution
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
NORMAL DISTRIBUTION (X)
4
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
3-5 Important Continuous Distributions
3-5.1 NORMAL DISTRIBUTION
Examples:
Properties of human, animal and plant populations (e.g., height, weight)
Test scores
Income levels
Errors in measurement
Compression strength of concrete
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
http://slidepdf.com/reader/full/ch3continuousdistributionsprobplotstorresgarcia02152016pdf 7/49 Reference: Most slides adapted from Montgomery et al. (2011)
NORMAL DISTRIBUTION
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
http://slidepdf.com/reader/full/ch3continuousdistributionsprobplotstorresgarcia02152016pdf 8/49 Reference: Most slides adapted from Montgomery et al. (2011)
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
![Page 9: Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf](https://reader034.fdocuments.net/reader034/viewer/2022052608/577c7b211a28abe054976086/html5/thumbnails/9.jpg)
8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
http://slidepdf.com/reader/full/ch3continuousdistributionsprobplotstorresgarcia02152016pdf 9/49 Reference: Most slides adapted from Montgomery et al. (2011)
3-5 Important Continuous Distributions
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
TRY THESE
10
When in doubt, draw a picture of thestandard normal density
Famous Quality Controlrelationship
Image source: mathisfun.com
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Reference: Most slides adapted from Montgomery et al. (2011)
NEXT EXAMPLE
11
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
NEXT EXAMPLE
12
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
CONFORMING EXAMPLE
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
CONFORMING EXAMPLE
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
REPRODUCTIVE PROPERTY OF THE NORMADISTRIBUTION
15
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE: SERIAL LINKAGE COMPO
16
= = 0.02 + 0.03 + 0.04 = 0.09 53.8 < < 54.2 = . −
. < < . −
.=
− < < = Φ 1.67 1 Φ 1.67 = 2Φ 1.67
1 = 2∗ 0.748 1 = 0.495
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE: SERIAL LINKAGE COMPO
17
= = 0.02 + 0.03 + 0.04 = 0.09 53.8 < < 54.2 = . −
. < < . −
.=
− < < = Φ 1.67 1 Φ 1.67 = 2Φ 1.67
1 = 2∗ 0.748 1 = 0.495
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
THE LOG NORMAL DISTRIBUTIONA variable is said to be distributed Log Normal if its natural logarithmis distributed Normally
Applications:Diameters of natural sediments
Onset of Alzheimer’s
Rainfall
Income of employed people
Permeability of some aquifers
Aquifer: An underground bed or layer of earth, gravel, or porous stone that yieldswater.
Low flow in some rivers
18
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
3-5 Important Continuous Distributions
3-5.2 Lognormal Distribution
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
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Reference: Most slides adapted from Montgomery et al. (2011)
Applications
Time between floods
Time between vehicles passing anobservation point
Time between earthquakes
Time to failure for electricalcomponents
Time between phone calls orbetween parts arriving at anassembly station
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
l is always a rate per unit. Units are often something/unit timeExample: l = 1/3 could mean 1 customer every 3 minutes
1/ l is always a unit. Unit is often time.
Example: 1/ l = 3 is the mean time between customers .
The Exponential Distribution
![Page 24: Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf](https://reader034.fdocuments.net/reader034/viewer/2022052608/577c7b211a28abe054976086/html5/thumbnails/24.jpg)
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Reference: Most slides adapted from Montgomery et al. (2011)
THE MEMORYLESS PROPERTY OF THE EXPDISTRIBUTION
0 1)()(
022
2
xe x X P x F
xe f(x) x
x
43
]41
1[
)]1(1[
41
43
41
41
41
and43
41
41
|1
43
223
)4/1(2
)1(2
X P eee
e
F
F
X P
X P
X P
X P X P
X X P
t X P s X P
t s X P s X t s X P |
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
0 301)( 30 for tet f
t
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8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf
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Reference: Most slides adapted from Montgomery et al. (2011)
0 )(1)(
0 1)(F(t)
0 30
1)(
30
30
30
for tet F t T P
for tet T P
for tet f
t
t
t A. What is the probability that abus will arrive in the next 60minutes?
P(T < 60) = F(60) = 1-e -2= .864
B. Given that a bus has not arrived in the past 60 minutes, what is theprobability that it will take longer than 90 minutes until one comes?
367.
306090
60
60and9060|90
30
30
12
3
3060
3090
e
T P ee
e
e
e
T P
T P
T P
T P T P T T P
Example
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Reference: Most slides adapted from Montgomery et al. (2011)
EXPONENTIAL DISTRIBUTION
We’ll talk a bit more about this distribution duringour Poisson Process discussion.
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
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Reference: Most slides adapted from Montgomery et al. (2011)
GAMMA SPECIAL CASES
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Reference: Most slides adapted from Montgomery et al. (2011)
3-5 Important Continuous Distributions
3-5.4 Weibull Distribution
= 1 −( / )
2
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
1 −( / )
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
1 −( / )
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Reference: Most slides adapted from Montgomery et al. (2011)
BETA DISTRIBUTION
Image source: Wikipedia.org
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
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Reference: Most slides adapted from Montgomery et al. (2011)
THE UNIFORM DISTRIBUTION
xba
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Reference: Most slides adapted from Montgomery et al. (2011)
The Uniform Distribution
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
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Reference: Most slides adapted from Montgomery et al. (2011)
EXAMPLE
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Reference: Most slides adapted from Montgomery et al. (2011)
T-DISTRIBUTION
Symmetric similar shape than Normal (in blue in picturebelow) but with heavier tails
42
Image Source: http://en.wikipedia.org/wiki/File:T_distribution_10df_enhanced.svg
2~and )1,0(~where k V N Z
k V
Z T
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Reference: Most slides adapted from Montgomery et al. (2011)
F-DISTRIBUTION
Testing for variance
43
Image Source: http://en.wikipedia.org/wiki/File:F_distributionPDF.png
22 ~and ~where/
/
V W
Y
uW F
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Reference: Most slides adapted from Montgomery et al. (2011)Image source:http://www.math.wm.edu/~leemis/chart/UDR/UDR.html
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Reference: Most slides adapted from Montgomery et al. (2011)
RANDOM VARIABLES AND PROBABIDISTRIBUTIONS1. Random Numbers, Random Variables (RVs), Random Experiment (3-1 & 3-2)
2. Probability Laws (3-3)
3. Continuous RVs (3-4 & 3-5)CRVs Distributions: Normal, Lognormal, Gamma, Weibull, Beta, and others
4. Probability Plots (3-6)
5. Discrete RVs (3-7 & 3-8)DRVs Distributions: Binomial
6. Poisson Process (3-9)
7. Normal Approximation to the Binomial and Poisson Distributions (3-10)
8. More than one RV and Independence (3-11)
9. Functions of RVs (3-12)
10. Statistics and the Central Limit Theorem (3-13)
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Reference: Most slides adapted from Montgomery et al. (2011)46
PROBABILITY PLOTS
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Reference: Most slides adapted from Montgomery et al. (2011)
3-6 Probability Plots
3-6.1 Normal Probability Plots• How do we know if a normal distribution is a reasonablemodel for data?• Probability plotting is a graphical method for determining
whether sample data conform to a hypothesized distributionbased on a subjective visual examination of the data.• Probability plotting typically uses special graph paper, know
as probability paper , that has been designed for thehypothesized distribution. Probability paper is widely availablfor the normal, lognormal, Weibull, and various chi-square andgamma distributions.
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Reference: Most slides adapted from Montgomery et al. (2011)
3-6 Probability Plots
3-6.1 Normal Probability Plots
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3-6 Probability Plots
3-6.2 Other Probability Plots