Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf

8/16/2019 Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf

http://slidepdf.com/reader/full/ch3continuousdistributionsprobplotstorresgarcia02152016pdf 1/49

Reference: Most slides adapted from Montgomery et al. (2011)

CHAPTER 3:CONTINUOUS DISTRIBUTIONS

AND PROBABILITY PLOTS

Instructor:Wandaliz Torres-García, Ph. D.


ININ 5559ENGINEERING STATISTICS




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)




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




NORMAL DISTRIBUTION (X)

4





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


http://slidepdf.com/reader/full/ch3continuousdistributionsprobplotstorresgarcia02152016pdf 7/49 Reference: Most slides adapted from Montgomery et al. (2011)

NORMAL DISTRIBUTION




3-5.1 Normal Distribution




TRY THESE

10

When in doubt, draw a picture of thestandard normal density

Famous Quality Controlrelationship

Image source: mathisfun.com




NEXT EXAMPLE

11




NEXT EXAMPLE

12




CONFORMING EXAMPLE




REPRODUCTIVE PROPERTY OF THE NORMADISTRIBUTION

15




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




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




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





3-5.2 Lognormal Distribution




EXAMPLE




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




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




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 |




EXAMPLE

0 301)( 30 for tet f

t




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




EXPONENTIAL DISTRIBUTION

We’ll talk a bit more about this distribution duringour Poisson Process discussion.




EXAMPLE




GAMMA SPECIAL CASES





3-5.4 Weibull Distribution

= 1 −( / )

2




EXAMPLE

1 −( / )




BETA DISTRIBUTION

Image source: Wikipedia.org




EXAMPLE




THE UNIFORM DISTRIBUTION

xba




The Uniform Distribution




EXAMPLE




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




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

u



Reference: Most slides adapted from Montgomery et al. (2011)Image source:http://www.math.wm.edu/~leemis/chart/UDR/UDR.html




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)



Reference: Most slides adapted from Montgomery et al. (2011)46

PROBABILITY PLOTS




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.





3-6.1 Normal Probability Plots




3-6.2 Other Probability Plots

Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf

Documents

Transcript of Ch3_ContinuousDistributions_ProbPlots_torresgarcia_02152016.pdf