Probability Theory

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Probability Theory

• Instructor
• Assoc. Prof. Dr. Deshi Ye ( ??? )
• College of Computer Science
• Zhejiang University
• Email yedeshi_at_zju.edu.cn

Course homepage Http//www.cs.zju.edu.cn/people/y
edeshi/probability08/
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Outline

• Brief introduction to the course
• Syllabus, course policies and contents
• Introduction to probability and statistics
• History and importance
• Treatment of data
• Graphs Pareto Diagram, Dot Diagram, Box-plot
• Frequency distribution, Stem-and-leaf Displays

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Course information

• What is for?
• This course provides an elementary introduction
  to probability with applications.
• Topics include
• axioms of probability
• basic probability concepts and models (counting
  methods , conditional probability, Bayes
  theorem,et.)
• random variablesindependence
• discrete and continuous probability
  distributions
• calculate mathematical expectation and variance
• limit theory

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Course Goals

• Students at the end of course should be able to
  do the following
• 1) Understand the concepts and methods of
  probability theory
• 2) Contrast, evaluate, and implement simulations
  or experiments
• 3) Utilize Minitab program for analyzing data and
  summarizing

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Syllabus

• Prerequisite one year course in calculus
• Textbooks (required) Miller Freund's
  Probability and Statistics for Engineers (Seventh
  Edition), Richard A. Johnson. Publishing House of
  Electronics Industry or Pearson Education Press.
• Chapter 1-6 for Probability Theory, Chapter
  7-13 for the second semester (Mathematical
  Statistics).
• References 1) A First course in Probability
  (6th Ed), Sheldon Ross. China Statistics Press.
  2) Probability Statistics for Engineers
  Scientists (7th Ed), R.E. Walpole,R.H. Myers,
  S.L. Myers, K. Ye. Tshinghua or Pearson Education
  Press.

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Grading

• Grades for the course will be based on the
  following weighting1) Class attendance 10 2)
  Homework assignment 20 3) Unit quiz 24 (8,
  8, 8)4) Final exam 46

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Homework

• 1) You may collaborate on homework, but you must
  write your submitted work in your own words. All
  steps are required, this includes showing
  calculations, derivations, and proofs. Solutions
  will be posted on the class web site.
• 2)Assignments are due in class as noted in the
  syllabus and web page. Homework assignments
  turned in within 48 hours of the due dates will
  be penalized 20 even if the solutions are
  "correct". HW more than two days late may not get
  graded at all.

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Checking web page

• I am highly recommend that each student check
  this web page at least once a week for new
  announcements and homework assignments.

www.cs.zju.edu.cn/people/yedeshi/probability08 /so
ftware/MiniTAB14.iso
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Probability in CS

• Randomized algorithms
• Querying Theory
• Software testing
• Computer simulation and modeling

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Cow path
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Introduction

• Probability theory is devoted to the study of
  uncertainty and variability
• Probability quantifies how uncertain we are about
  future events
• Statistics can be described as the study of how
  to make inference and decisions in the face of
  uncertainty and variability

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Uncertainty Events

• Say red
• Coin toss
• Matching games (Cards, Name)
• Traffic light
• The life of a light
• Lotteries?

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Poker Lotteries

• http//www.zjlottery.com/news/showmes.asp?newsid9
  950
• Heart, Spade, Club, Diamonds
• 1(A)?2?3?4?5?6?7?8?9?10?11(J)?12(Q)?13(K)
• Arbitrarily choose one piece
• 2?,if win you are awarded 13? (win in 1/13)

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Why measure uncertainty?

• To make tradeoffs among uncertain events
• Measure combined effect of several uncertain
  events
• To communicate about uncertainty

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Brief History

• Blaise Pascal and Pierre de Fermat the origins
  of probability are found.
• concerning a popular dice game
• fundamental principles of probability theory
• Pierre de Laplace
• Before him, concern on the analysis of games of
  chance
• Laplace applied probabilistic ideas to many
  scientific and practical problems

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History cont.

• Mathematical statistics is one important branch
  of applied probability other applications occur
  in such widely different fields as genetics,
  psychology, economics, engineering, computer
  science.
• Important workers Chebyshev, Markov, von Mises,
  and Kolmogorov
• One of the difficulties is the definition of
  probability. 20th century, it was solved by
  treating probability theory on an axiomatic basis
  (Kolmogorov).

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Words for probability

• Chance the falling out or happening of events
• Stochastic randomly determined
• Random not sent or guided in a special
  direction, having no definite aim or purpose
• Aleatory dependent on the throw of a die
• Hazard a chance or venture.

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Importance of Prob. Theory

• Two major applications of Prob.
• Risk assessment (new medical treatments)
• Reliability (weather prediction, earthquake,
  reduce failure of consumer product)
• Why statistics and probability in engineering?
• Quantify the uncertainty associated with engineer
  model
• Evaluate the result of experiment
• Assess importance of measurement uncertainty
• Safeguard for persons, qualities of environment,
  assets

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A case study

• Visually inspecting data to improve product
  quality
• Monitoring manufacturing data
• Ceramic part in coffee makers,
• which is made by filling the
• mixture of clay-water-oil.
• The depth of the slot is uncontrolled.
• Slot depth was measured on three ceramic parts
  selected from production every half hour during
  the first 6 AM to 3 PM.

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Time series plot
Stable 217.5 Good quality 215, 220
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Ch2 Treatment of data

• Outline
• Pareto diagrams, dot diagrams
• Histograms (Frequency distributions)
• Stem-and-leaf display
• Box-plot (Quartiles and Percentiles)
• The calculation of mean and standard
  deviation s

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Pareto Diagram

• Pareto Diagram display orders each type of
  failure or defect according to its frequency.
• For a computer-controlled lathe whose performance
  was below par, workers recorded the following
• causes and their frequencies
• power fluctuations 6
• controller not stable 22
• operator error 13
• worn tool not replaced 2
• other 5

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Minitab14

• 1. Stat-gtQuality tools-gtPareto chart
• 2. Choose chart defects table as follows

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Output
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Pareto diagram

• Pareto diagram depicts Paretos empirical law
  that any assortment of events consists of a few
  major and many minor elements.
• Typically, two or three elements will account for
  more than half of the total frequency, i.e., it
  points out the main causes.

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Pareto diagram--application

• Software testing
• Software defect distribution

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Dot diagram

• Second step to improve the quality of lathe,
• Data were collected from observation on the
  deviations of cutting speed from the target value
  set by the controller.
• EX. Cutting speed target speed
• 3 6 2 4 7 4
• Dot diagram A number line in which one dot is
  placed above a value on the number line for each
  occurrence of that value. That is, one dot means
  the value occurred once, three dots mean the
  value occurred three times, etc.
• In minitab stat-gtdotplots-gtsimple

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Dot diagram

• This diagram visually summarize the information
  that the lathe is generally running fast.

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Multiple sample

• C1 0.27 0.35 0.37
• C2 0.23 0.15 0.25 0.24 0.30 0.33 0.26

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Frequency distributions

• A frequency distribution is a tabular arrangement
  of data whereby the data is grouped into
  different intervals, and then the number of
  observations that belong to each interval is
  determined.
• Data that is presented in this manner are known
  as grouped data.

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Data001. 80 data of emission (in ton)of sulfur
oxides from an industry plant

• 15.8 26.4 17.3 11.2 23.9 24.8 18.7 13.9 9.0 13.2
  22.7 9.8 6.2 14.7 17.5 26.1 12.8 28.6 17.6 23.7
  26.8
• 22.7 18.0 20.5 11.0 20.9 15.5 19.4 16.7 10.7
  19.1 15.2 22.9 26.6 20.4 21.4 19.2 21.6 16.9
  19.0 18.5 23.0
• 24.6 20.1 16.2 18.0 7.7 13.5 23.5 14.5 14.4
  29.6 19.4 17.0 20.8 24.3 22.5 24.6 18.4 18.1 8.3
  21.9 12.3
• 22.3 13.3 11.8 19.3 20.0 25.7 31.8 25.9 10.5
  15.9 27.5 18.1 17.9 9.4 24.1 20.1 28.5

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Class limits frequnecy
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Class limit and width

• lower class limit The smallest value that can
  belong to a given interval
• upper class limit The largest value that can
  belong to the interval.
• Class width The difference between the upper
  class limit and the lower class limit is defined
  to be the class width.

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Guidelines for classes

• 1. There should be between 5 and 20 classes.
• 2.The class width should be an odd number. This
  will guarantee that the class midpoints are
  integers instead of decimals.
• 3. The classes must be mutually exclusive. This
  means that no data value can fall into two
  different classes
• 4. The classes must be all inclusive or
  exhaustive. This means that all data values must
  be included.
• 5. The classes must be continuous. There are no
  gaps in a frequency distribution. Classes that
  have no values in them must be included (unless
  it's the first or last class which are dropped).
• 6.The classes must be equal in width. The
  exception here is the first or last class. It is
  possible to have an "below ..." or "... and
  above" class. This is often used with ages

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Steps

• 1. Find the largest and smallest values
• 2. Compute the Range Maximum - Minimum
• 3. Select the number of classes desired. This is
  usually between 5 and 20.
• 4. Find the class width by dividing the range by
  the number of classes and rounding up.

You must round up, not off. Normally 3.2 would
round to be 3, but in rounding up, it becomes 4.
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Class limits frequnecy
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Variants of frequency distribution

• The cumulative frequency distribution is obtained
  by computing the cumulative frequency, defined as
  the total frequency of all values less than the
  upper class limit of a particular interval, for
  all intervals.
• Relative frequency the ratio of the number of
  observations in the interval to the total number
  of observations
• The percentage frequency distribution is arrived
  at by multiplying the relative frequencies of
  each interval by 100.

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Cumulative frequency
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Percentage distribution
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Histogram

• The most common form of graphical presentation of
  a frequency distribution is the histogram.
• Histogram is constructed of adjacent rectangles
  the height of the rectangles is the class
  frequencies and the bases of the rectangles
  extend between successive class boundaries.

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Histogram in Minitab
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Histogram in Minitab

• Graph-gthistogram-gtsimple
• Graph variables c4 (all data in a column)
• Edit bars Click the bars in the output figures,
  in Binning, Interval type select midpoint and
  interval definition select midpoint/cutpoint, and
  then input 7 11 15 19 23 27 31 as illustrated in
  the following

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Density histogram

• When a histogram is constructed from a frequency
  table having classes of unequal lengths, the
  height of each rectangle must be changed to
• Height relative frequency / width.
• The area of the rectangle then represents the
  relative frequency for the class and the total
  area of the histogram is 1.

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Density histogram
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Density Histogram

• Graph-gthistogram-gtsimple
• Scale-gtY-Scale Type-gtDensity
• Edit Bars-gtBinning-gtCut point-gt
• 5 13 17 21 25 29 33

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Cumulative histogram

• 1) Graph-gthistogram-gtsimple
• 2) Dataview-gt
• Datadisplay check symbos only
• Smoother check lowess and 0 in degree of
  smoothing and 1 in number of steps.

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Stem-and-leaf Display

• Class limits and frequency, contain data in each
  class, but the original data points have been
  lost.
• Stem-and-leaf A data plot which uses part of the
  data value as the stem and the rest of the data
  value (the leaf) to form groups or classes. This
  is very useful for sorting data quickly.
• Stem-and-leaf function the same as histogram but
  save the original data points.
• Example 11 numbers
• 12, 13, 21, 27, 33, 34, 35, 37, 40, 40, 41

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• Frequency table
• Class limits Frequency
• 10 19 2
• 20 29 2
• 30 39 4
• 40 49 3

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Stem-and-leaf
Stem-and-leaf each row has a stem and each digit
on a stem to the right of the vertical line is a
life. The "stem" is the left-hand column which
contains the tens digits. The "leaves" are the
lists in the right-hand column, showing all the
ones digits for each of the tens, twenties,
thirties, and forties. Key 40 means 40
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Stem-and-leaf Display

• Example in P23 20 numbers
• 29, 44, 12, 53, 21, 34, 39, 25, 48, 23
• 17, 24, 27, 32, 34, 15, 42, 21, 28, 27

Frequency table Class limits Frequency
10 19 3 20 29 9
30 39 4 40 49 3
50 59 1
Stem-and-leaf 1 2 5 7 2 1 1 3 4 5 7 7 8
9 3 2 4 4 9 4 2 4 8 5 3
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Stem-and-leaf in Minitab

• The display has three columns
• The leaves (right) - Each value in the leaf
  column represents a digit from one observation.
• The stem (middle) - The stem value represents the
  digit immediately to the left of the leaf digit.
• Counts (left) - If the median value for the
  sample is included in a row, the count for that
  row is enclosed in parentheses. The values for
  rows above and below the median are cumulative.

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Stem-and-leaf for DATA001

• Stem-and-leaf of frequencies N 80
• Leaf Unit 1.0
• 2 0 67
• 6 0 8999
• 11 1 00111
• 17 1 223333
• 24 1 4445555
• 32 1 66677777
• (13) 1 8888888999999
• 35 2 0000000111
• 25 2 222223333
• 16 2 4444455
• 9 2 66667
• 4 2 889
• 1 3 1

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Ch2.5 Descriptive measures

• Mean the sum of the observation divided by the
  sample size.
• Median the center, or location, of a set of
  data. If the observations are arranged in an
  ascending or descending order
• If the number of observations is odd, the median
  is the middle value.
• If the number of observations is even, the median
  is the average of the two middle values.

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Example

• 15 14 2 27 13
• Mean
• Ordering the data from smallest to largest
• 2 13 14 15 27
• The median is the third largest value 14

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Other central tendency

• Midrange
• The midrange is simply the midpoint between the
  highest and lowest values.
• Mode
• The mode is the most frequent data value. There
  may be no mode if no one value appears more than
  any other. There may also be two modes (bimodal),
  three modes (trimodal), or more than three modes
  (multi-modal).

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Summary

• The Mean is used in computing other statistics
  (such as the variance) and does not exist for
  open ended grouped frequency distributions. It is
  often not appropriate for skewed distributions
  such as salary information.
• The Median is the center number and is good for
  skewed distributions because it is resistant to
  change.
• The Mode is used to describe the most typical
  case. The mode can be used with nominal data
  whereas the others can't. The mode may or may not
  exist and there may be more than one value for
  the mode
• The Midrange is not used very often. It is a very
  rough estimate of the average and is greatly
  affected by extreme values (even more so than the
  mean).

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Summary cont.
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Sample variance

• Deviations from the mean
• Standard deviation s

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Quartiles and Percentiles

• Quartiles are values in a given set of
  observations that divide the data in 4 equal
  parts.
• The first quartile, , is a value that has one
  fourth, or 25, of the observation below its
  value.
• The sample 100 p-th percentile is a value such
  that at least 100p of the observation are at or
  below this value, and at least 100(1-p) are at
  or above this value.

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Example

• Example in P34

N/4 is an integer, take the average Or round up,
otherwise
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Boxplots

• A boxplot is a way of summarizing information
  contained in the quartiles (or on a interval)
• Box length interquartile range

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Quartile calculation in Minitab

• The first quartile (Q1) is the observation at
  position (n1) / 4, and the third quartile (Q3)
  is the observation at position 3(n1) / 4, where
  n is the number of observations. If the position
  is not an integer, interpolation is used.
• For example, suppose n10. Then (10 1)/4
  2.75, and Q1 is between the second and third
  observations (call them x2 and x3), three-fourths
  of the way up. Thus, Q1 x2 0.75(x3 - x2).
  Since 3(10 1)/4 8.25, Q3 x8 0.25(x9 -
  x8), where x8 and x9 are the eight and ninth
  observations.
• Indeed, Choose Hinges in BoxEndpoints, will get
  Quartile as in Textbook.

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Modified boxplot
Upper limit Q3 1.5 (Q3 - Q1)

• Outlier too far from third quartile.
• Largest observation within 1.5(interquartile
  range) of third quartile.
• Modified boxplot identify outliers and reduce
  the effect on the shape of the boxplot.

Lower limit Q1- 1.5 (Q3 - Q1)
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Homework 1

• Problems in Textbook (2.63,2.72,2.76)  
• Due date Sept 16.

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Conclusion

• Graph the data as a dot diagram or histogram or
  box plot to assess the overall pattern of data
• Group the data by frequency distribution,
  Stem-and-leaf
• Calculate the summary statistics-sample mean,
  standard deviation, and quartiles to describe
  the data set.

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The END
Thanks !
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Population and Sample

• Investigating a physical phenomenon, production
  process, or manufactured unit, share some common
  characteristics.
• Relevant data must be collected.
• Unit the source of each measurement.
• A single entity, usually an object or person
• Population entire collection of units.

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Examples
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Sample

• Statistical population the set of all
  measurement corresponding to each unit in the
  entire population of units about which
  information is sought.
• Sample A sample from a statistical population is
  the subset of measurements that are actually
  collected in the course of investigation.

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Sample

• Need to be representative of the population
• To be large enough to contain sufficient
  information to answer the question about the
  population

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Discussion

• P10, Review Exercises 1.2
• A radio-show host announced that she wanted to
  know which singer was the favorite among college
  students in your school. Listeners were asked to
  call and name their favorite singer. Identify the
  population, in terms of preferences, and the
  sample.
• Is the sample likely to be more representative?
• Comment. Also describe how to obtain a sample
  that is likely to be more representative.