Probability Lecture ONE

8/8/2019 Probability Lecture ONE

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Probability & Statistics forBME

(Third Year)2009-2010

Assistant Prof \

Fadhl M. Alakwaa


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This course provides the foundational skills for

advanced statistical analysis of biological signals.The

basic analytical concepts of probability theory,

statistical design of experiments and data analysis andrepresentation of biological variables as random

processes are demonstrated and practiced through

computer based analysis of biologically relevant data

sets provided throughout the course. All computationalhomework assignments are carried out using

MATLAB.

Course Description


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Solve simple probability problems with electrical and computer

engineering applications using the basic axioms of probability.

Describe the fundamental properties of probability density

functions with applications to single and multivariate randomvariables.

Describe the functional characteristics of probability density

functions frequently encountered in life science research such as

the Binomial, Uniform, Gaussian and Poisson.

Determine the first through fourth moments of any probabilitydensity function using the moment generating function.

Calculate confidence intervals and levels of statistical

significance using fundamental measures of expectation and

variance for a given numerical data set.

Course Objective

Upon completion of the course, students should be able to


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Discern between random variables and random processes for

given mathematical functions and numerical data sets.

Determine the power spectral density of a random process for

given mathematical functions and biological data sets.

Determine whether a random process is ergodic or nonergodicand demonstrate an ability to quantify the level of correlation

between sets of random processes for given mathematical

functions and biological data sets.

Model complex families of signals by means of random

processes. Determine the random process model for the output of a linear

system when the system and input random process models are

known.

Course Objective

Upon completion of the course, students should be able to


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1. Intuitive Probability and Random Processes using MATLAB

by Steven Kay (2006), Springer. ISBN: 0-387-24157-4.

2. Applied Statistics and Probability for Engineers by

Montgomery, Runger 4th Edition

3. Probability and Statistics for Engineering and the Sciences

by Jay L. Devore 6th Edition 2004 ISBN: 0534399339

References


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Essential Book


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Recommended Book


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CourseL

abMatlab

Excel

Minitab

labview


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Course Evaluation


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Course Projects: Project1


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Course Projects: Project2

Understand Manual service for medical

instrument


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Course Presentation

Bayes theorem

Probability mass function PMF

Binomial distribution

Hypergometric distribution

Poisson distribution

Normal distribution

Joint probability distribution

Marginal probability distribution

Conditional probability distributionCovariance and correlation

Box plots

Time sequence plots

Histograms

Central limit thermo

Point estimation of parameters

Maximum likelihood

Confidence interval

t distribution

Hypothesis testing

P-value

Type I and type II error

Statistical inference

F distribution

Least square estimate

Linear regression

Design engineering experiment

ANOVA

Cumulative distribution functions

Means and variance of discrete variable

Means and variance of continuous variable

Choose one topics from the below and do a power point

presentation:


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Important comments from the

previous course Not Excuses

Not degree explanation (fair assessment)

In time policy (one day late=one degree

loss)

Join a group (mandatory)

Update your attendance and results daily.


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Group Activity: BME_UST

http://www.facebook.com/search/?q=BME_UST&init=quick#!/grou

p.php?gid=325135515239&ref=search&sid=1096082202.17723631

20..1


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An engineeris someone who solves problems of

interest to society by the efficient application ofscientific principles by

Refining existing products

Designing new products or processes

1-1 The Engineering Method and

Statistical Thinking


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Figure 1.1 The engineering method

1- Develop a clear and concise description of the problem

2- Identify, the important factors that affect this problem3- Propose a model for the problem using scientific or

engineering knowledge of the phenomenon being studied. State

any limitations or assumptions of the model


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Statistical Thinking..continue

4- Conduct appropriate experiments and collect data to test or

validate the model.

5- Refine the model on the basis of the observed data.

6- Manipulate the model to assist in developing solution to the

problem.7- Conduct an appropriate experiment to confirm that the

proposed solution to the problem is both effective and efficient.

8- Draw conclusions or make recommendation based on the

problem solution.


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The field ofstatistics deals with the collection,

presentation, analysis, and use of data to

Make decisions

Solve problems

Design products and processes




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Statistical techniques are useful for describing and

understanding variability.

By variability, we mean successive observations of asystem or phenomenon do notproduce exactly the same

result.

Statistics gives us a framework for describing thisvariability and for learning about potential sources of

variability.




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Engineering Example

An engineer is designing a nylon connector to be used in an

automotive engine application. The engineer is consideringestablishing the design specification on wall thickness at 3/32

inch but is somewhat uncertain about the effect of this decision

on the connector pull-off force. If the pull-off force is too low, the

connector may fail when it is installed in an engine. Eightprototype units are produced and their pull-off forces measured

(in pounds): 12.6, 12.9, 13.4, 12.3, 13.6, 13.5, 12.6, 13.1.




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Engineering Example

The dot diagram is a very useful plot for displaying a small

body of data - say up to about 20 observations.

T

his plot allows us to see easily two features of the data; thelocation, or the middle, and the scatterorvariability.



3/32 inch


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Engineering Example

The engineer considers an alternate design and eight prototypes

are built and pull-off force measured.

T

he dot diagram can be used to compare two sets of data

Figure 1-3 Dot diagram of pull-off force for two

wall thicknesses.



average13.0

13.4


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How do you know that another sample of prototypes will not

give different results?

If we use the test results obtained so far to conclude that

increasing the wall thickness increases the strength, what risks

are associated with this decision?

For example, is it possible that the apparent increase in

pull-off force observed in the thicker prototypes is only

due to the inherent variability in the system and that

increasing the thickness of the part (cost) really has no

effect on the pull-off force?




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Engineering Example

Since pull-off force varies or exhibits variability, it is a

random variable.

A random variable, X, can be model by

X = Q + I

where Q is a constant and I a random disturbance.

Due to environment, test equipment, different in

individual parts themselves.




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8 connectors

prototypes

Connectors that

will be sold to

customers

Sampling error, sample size


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Three basic methods for collecting data:

A retrospective study using

historical data

An observational study

A designed experiment

1-2 Collecting Engineering Data


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1-2.2 Retrospective Study

The retrospective study may involve a lot of data, but

that data may contain relatively little useful information

about the problem.

Some of the relevant data may be missing

There may be recording errors resulting in outliers

(unusual values)

Other important factors may not have been collected

1-2.3 Observational Study


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1-2.4 Designed Experiments


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Figure 1-5 The factorial design for the distillation column


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Figure 1-6 A four-factorial experiment for the distillation column


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1-2.5 Observing Processes Over Time

Whenever data are collected over time it is important to plot

the data over time. Phenomena that might affect the system

or process often become more visible in a time-oriented plot

and the concept of stability can be better judged.

Figure 1-8 The dot diagram illustrates variation but doesnot identify the problem.


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Figure 1-9 A time series plot of concentration provides

more information than a dot diagram.


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Figure 1-10 Demings funnel experiment.


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Figure 1-11 Adjustments applied to random disturbancesover control the process and increase the deviations from

the target.


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Figure 1-12 Process mean shift is detected at observationnumber 57, and one adjustment (a decrease of two units)

reduces the deviations from target.


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Figure 1-13 A control chart for the chemical process

concentration data.


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Figure 1-14 Enumerative versus analytic study.


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1-3 Mechanistic and Empirical Models

A mechanistic model is built from our underlying

knowledge of the basic physical mechanism that relates

several variables.

Example: Ohms Law

Current = voltage/resistance

I=

E/R

I=E/R + I


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An empirical model is built from our engineering and

scientific knowledge of the phenomenon, but is not

directly developed from our theoretical or first-

principles understanding of the underlying mechanism.


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Example

Suppose we are interested in the number averagemolecular weight (Mn) of a polymer. Now we know that Mn

is related to the viscosity of the material (V), and it alsodepends on the amount of catalyst (C) and the temperature(T) in the polymerization reactor when the material ismanufactured. The relationship between Mn and thesevariables is

Mn = f(V,C,T)say, where the form of the function fis unknown.

where the Fs are unknown parameters.


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In general, this type of empirical model is called a

regression model.

The estimated regression line is given by


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Figure 1-15 Three-dimensional plot of the wire and pullstrength data.


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Figure 1-16 Plot of the predicted values of pull strengthfrom the empirical model.


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1-4 Probability and Probability Models

Probability models help quantify the

risks involved in statistical inference, that

is, risks involved in decisions made everyday.

Probability provides the frameworkfor

the study and application of statistics.


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Documents

Probability Lecture ONE