MATH408: Probability

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MATH408 Probability StatisticsSummer
1999WEEK 2
Dr. Srinivas R. Chakravarthy Professor of
Mathematics and Statistics Kettering
University (GMI Engineering Management
Institute) Flint, MI 48504-4898 Phone
810.762.7906 Email schakrav_at_kettering.edu Homepag
e www.kettering.edu/schakrav
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SAMPLE

• Sample is a subset (part) of the population.
• Since it is infeasible (and impossible in many
  cases) to study the entire population, one has to
  rely on samples to make the study.
• Samples have to be as representative as possible
  in order to make valid conclusions about the
  populations under study.

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SAMPLE (cont'd)

• Contain more or less the same type of information
  that the population has.
• For example if workers from three shifts are
  involved in assembling cars of a particular
  model, then the sample should contain units from
  all three.
• Samples will be used to estimate the parameters.

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SAMPLE (contd)

• Much care should be devoted to the sampling.
• There is always going to be some error involved
  in making inferences about the populations based
  on the samples.
• The goal is to minimize this error as much as
  possible.
• There are many ways of bringing in systematic
  bias (consistently misrepresent the population).

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SAMPLE (contd)

• This can be avoided by taking random samples.
• Simple random sample all units are equally
  likely to be selected.
• Multi-stage sample units are selected in several
  stages.
• Cluster sample is used when there is no list of
  all the elements in the population and the
  elements are clustered in larger units.

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SAMPLE (contd)

• Stratified sample In cases where population
  under study may be viewed as comprising different
  groups (stratas) and where elements in each group
  are more or less homogeneous, we randomly select
  elements from every one of the strata.
• Convenience sample samples are taken based on
  convenience of the experimenter.
• Systematic sample units are taken in a
  systematic way such as selecting every 10th item
  after selecting the first item at random.

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HOW TO USE SAMPLES?

• Samples should represent the population.
• Random sample obtained will not always be an
  exact copy of the population.
• Thus, there is bound to be some error

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SAMPLES (contd)

• Random or unbiased error This is due to the
  random selection of the sample and the mean of
  such error will be 0 as positive deviation and
  negative deviation cancel out. This random error
  is also referred to as random deviation and is
  measured by the standard deviation of the
  estimator.
• Non-random or biased error this occurs due to
  several sources such as human, machines, mistakes
  due to copying or punching, recording and so on.
  Through careful planning we should try to avoid
  or minimize this error.

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EXPERIMENTS USING MINITAB

• We will illustrate the concepts of sample,
  sampling error, etc with practical data using
  MINITAB when we go to the laboratory next time.

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NEXT?

• Once the data has been gathered, what do we do
  next?
• Before any formal statistical inference through
  estimation or test of hypotheses is conducted,
  EDA should be employed.

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EXPLORATORY DATA ANALYSIS (EDA)

• This is a procedure by which the data is
  carefully looked for patterns, if any, and to
  isolate them.
• First step in identifying appropriate model.

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EDA(cont'd)

• The main difference between EDAand conventional
  data analysis is
• while the former, which is more flexible (in
  terms of any assumptions on the nature of the
  populations from which the data are gathered)
  emphasizes on searching for evidence and clues
  for the patterns, the latter concentrates on
  evaluating the evidence and the hypotheses on the
  nature of the parameters of the population(s)
  under study.

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CAPABILITY ANALYSIS

• Deals with the study of the ability of the
  process to manufacture products within
  specifications.
• In order to perform the capability analysis, the
  process must be stable (i.e., things such as warm
  up period needed on the process before
  manufacturing products and others should be taken
  care of).

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CAPABILITY ANALYSIS (cont'd)

• The process specifications are compared to the
  variance (or the spread) of the process.
• For a process to be more capable, more
  measurements would be expected to fall within the
  specifications.

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CAPABILITY ANALYSIS (cont'd)
A commonly used capability index is given by
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CAPABILITY ANALYSIS (cont'd)

• The larger the value of Cpk, the less evidence
  that the process is outside the specifications. A
  value of 1.5 or higher for Cpk is usually
  desired. More on this will be seen later.

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DESCRIPTIVE STATISTICS

• Deals with characterization and summary of key
  observations from the data.
• Quantitative measures mean, median, mode,
  standard deviation, percentiles, etc.
• Graphs histogram, Box plot, scatter plot, Pareto
  diagram, stem-and-leaf plot, etc.
• Here one has to be careful in interpreting the
  numbers. Usually more than one descriptive
  measure will be used to assess the problem on
  hand.

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DIFFERENT TYPES OF PLOTS

• Point plot The horizontal axis (x-axis) covering
  the range of the data values and vertically plot
  the points, stacking any repeated values.
• Time series plot x-axis corresponds to the
  number of the observation or the time of the
  observation or the day and so on and the y-axis
  will correspond to the value of the observation.

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Time-series plot
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PLOTS (cont'd)

• Scatter plot Construct x-axis and y-axis that
  cover the ranges of two variables. Plot (xi, yi)
  points for each observation in the data set.
• Histogrom This is a bar graph, where the data is
  grouped into many classes. The x-axis corresponds
  to the classes and the y-axis gives the frequency
  of the observations.

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Histogram
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PLOTS (cont'd)

• Stem-and-leaf plot Data is plotted in such a way
  the output will look like histogram and also
  features a frequency distribution. The idea is to
  use the digits of the data to illustrate its
  range, shape and density. Each observation is
  split into leading digits and trailing digits.
  All the leading digits are sorted and listed to
  the left of a vertical line. The trailing digits
  are written to the right of the vertical line.

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Stem-and-leaf
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PLOTS (cont'd)

• Pareto Diagram Named after the Italian
  economist. This is a bar diagram for qualitative
  factors. This is very useful to identify and
  separate the commonly occurring factors from the
  less important ones. Visually it conveys the
  information very easily.

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Pareto Diagram
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PLOTS (cont'd)

• Box plot is due to J. Tukey and provides a great
  deal of information. A rectangle whose lower and
  upper limits are the first and third quartiles,
  respectively, is drawn. The median is given by a
  horizontal line segment inside the rectangle box.
  The average value is marked by a symbol such as
  x or . All points that are more extreme are
  identified.

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Boxplot for MPG example
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PLOTS (cont'd)

• Quantile plot This plot is very useful when we
  want to identify/ verify an hypothesized
  population distribution from which the data set
  could have been chosen. A quantile, Q(r), is a
  number that divides a sample (or population) into
  two groups so that the specified fraction r of
  the data values is less than or equal to the
  value of the quantile.

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PLOTS (cont'd)

• Probability plot This involves plotting the
  cumulative probability and the observed value of
  the variable against a suitable probability scale
  which will result in linearization of the data.
  The basic steps involved here are (a) Sorting
  the data into ascending order (b) Computing the
  plotting points (c) Selecting appropriate
  probability paper (d) Plot the points (e)
  Fitting a best line to data.

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MEASURES OF LOCATION

• MEAN Used very often in analyzing the data.
• Although this is a common measure, if the data
  vary greatly the average may take a non-typical
  value and could be misleading.
• Median is the halfway point of the data and
  tells us something about the location of the
  distribution of the data.
• Mode if exists, gives the data point that occur
  most frequently.
• It is possible for a set of data to have 0, 1 or
  more modes.

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LOCATION (contd)

• Mean and median always exist.
• Mode need not exist.
• Median and mode are less sensitive to extreme
  observations.
• Mean is most widely used.
• There are some data set for which median or mode
  may be more appropriate than mean

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LOCATION (contd)

• Percentiles The 100pth percentile of a set of
  data is the value below which a proportion p of
  the data points will lie.
• Percentiles convey more information and are very
  useful in setting up warranty or guarantee
  periods for manufactured items.
• Also referred to as quantiles.
• The shape of the frequency data can be classified
  into several classes.

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LOCATION (contd)

• Symmetric mean median mode
• Positively skewed tail to the right mean gt
  median
• Negatively skewedtail to the right median gt
  mean
• In problems, such as waiting time problems one is
  interested in the tails of the distributions.
• For skewed data median is preferred to the mean.

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MEASURES OF SPREAD

• One should not solely rely on mean or median or
  mode.
• Also two or more sets of data may have the same
  mean but they may be qualitatively different.
• In order to make a meaningful study, we need to
  rely on other measures.

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MEASURES OF SPREAD

• For example, we may be interested to see how the
  data is spread.
• Range is the difference between the largest and
  the smallest observations.
• Quick estimate on the standard deviation.
• Plays an important role in SPC.

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SPREAD (contd)

• Standard deviation describes how the data is
  spread around its mean.
• Coefficient of variation The measures we have
  seen so far depend on the unit of measurements.
  It is sometimes necessary and convenient to have
  a measure that is independent of the unit and
  such a useful and common measure is given by the
  ratio of the standard deviation to the mean
  called the coefficient of variation.

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SPREAD (contd)

• Interquartile range is the difference between
  the 75th and 25th percentiles.
• Gives the interval which contains the central 50
  of the observations.
• Avoids the total dependence on extreme data

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Stem-and-leaf of cycles N 70 Leaf Unit
100 (Problem 2.2) 1 0 3 1
0 5 0 7777 10 0 88899 22 1
000000011111 33 1 22222223333 (15) 1
444445555555555 22 1 66667777777 11 1
888899 5 2 011 2 2 22
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INFERENTIAL STATISTICS

• Recall that a parameter is a descriptive measure
  of some characteristic of the population.
• The standard ones are the mean, variance and
  proportion.
• We will simply denote by ?, the parameter of the
  population under study.

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INFERENTIAL STATISTICS

• Estimation Theory and Tests of Hypotheses are two
  pillars of statistical inference.
• While estimation theory is concerned about giving
  point and interval estimates for parameter(s)
  under study, test of hypotheses deals with
  testing claims on the parameter(s).

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Illustrative Example 1

• The following data corresponds to an experiment
  in which the effect of engine RPM on the
  horsepower is under study.
• TABLE 1 Data for HP Example

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GROUNDWORK FOR PROBABILITY

• Looking at the data in Table 1, why is that the
  hp values, say at 4500 RPM, are not exactly the
  same if the experiment is repeated under the
  same conditions?
• The fluctuation that occurs from one repetition
  to another is called experimental variation,
  which is usually referred to as noise or
  statistical error or simply error Recall
  this term from earlier discussion on data
  collection.

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PROBABILITY (contd)

• This represents the variation that is
  inherently present in any (practical) system.
• The noise is a random variable and is studied
  through probability.

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What is Probability?

• A manufacturer of blender motors wants to
  determine the warranty period for this product.
• If motor life were constant, (say 8 years) the
  manufacturer would have no problem. The motor
  could be warranted for 8 years.
• But, in reality, the motor life is not a constant.

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PROBABILITY (contd)

• Some motors will fail quickly and others will
  last for several years.
• There is an element of randomness in the life of
  the motors.
• The manufacturer cannot precisely predict how
  long any motor will last.
• Probability theory gives the manufacturer the
  means to quantify what is known about motor
  lifetimes and helps to quantify the risks
  involved in setting a warranty period.

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PROBABILITY (contd)

• Similar problems arise in the context of other
  products.
• FMS play an important role in modern
  manufacturing. Improved quality, lower inventory,
  shorter lead times, higher productivity and
  greater safety are some of the benefits derived
  from FMS.
• All of these have random elements.

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PROBABILITY (contd)

• Probability theory deals with randomness,
  allowing the study of quantities whose behavior
  cannot be predicted completely in advance.
• The above examples deal with manufacturing.

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PROBABILITY (contd)

• We could just as easily find examples in
  business, electrical and computer engineering,
  biomedical science and engineering, sociology,
  economics, marketing, civil engineering, the
  behavioral sciences and so on. The underlying
  problem, randomness, is the same.

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PROBABILITY (contd)

• One should understand the ideas of probability
  and statistics from both theoretical and
  practical points of view.
• To properly apply probability and statistics in
  the real world, we must appreciate both sides of
  the picture.
• We cannot properly apply a procedure if we don't,
  at least in general terms, understand the
  reasoning (theory) behind it.

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PROBABILITY (contd)

• On the other hand, trying to apply theory without
  knowledge of the area of application is foolish.
  We have to have a proper perspective on both
  before meaningful progress can be made.
• Probability theory develops mathematical models
  for random experiments.
• A random experiment is a sequence of actions
  whose outcome cannot be predicted with certainty.

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PROBABILITY (contd)

• If you've used phrases like "one chance in a
  1000", "50-50" or "3-to-2 odds" to describe
  something, you have most likely been using an
  informal probability model.
• If we throw two fair dice and our concern is
  about whether or not the dice eventually land and
  come to rest, then the throwing of the two dice
  is not a random experiment.
• Our knowledge of physical laws allows us to
  predict with virtual certainty that this outcome
  will happen.

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PROBABILITY (contd)

• If, however, we are concerned with how many dots
  show on the topmost faces when the dice come to
  rest, then we are performing a random experiment
  in tossing the dice, since we cannot predict with
  certainty which faces will show.

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PROBABILITY (contd)

• Outcomes of random experiments the length of a
  phone call, the gender mix of three people chosen
  from a group of 25 people, and the phenotype of
  the offspring of a cross breeding experiment, the
  number of defects on a painted panel.

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EXPERIMENT

• Calculation of MPG of a new model car.
• Measurements of current in a thin copper wire.
• Measurements of Film build thickness in a
  painting process.
• Duration of phone calls.
• Time to assemble a job.
• Tossing a coin.
• Throwing a dice.

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Sample space (S)

• Collection of all possible outcomes in an
  experiment.
• The MPGs of all cars from that particular model
  car.
• Event (A)
• A subset of a sample space
• The MPG of the new model car exceeds, say, 25
  miles.

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SET THEORY
A?B
A?B
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SET THEORY (contd)
A'
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PROBABILITY

• is a function defined on the set of all possible
  events.
• is a number between 0 and 1.
• satisfies a set of axioms
• P(A) ? 0.
• P(S) 1.

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PROPERTIES

• 0 ? P(A) ?1.
• P(A') 1 - P(A)
• P(A?B) P(A) P(B) - P(A?B).

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Classical definition

• While axiomatic definiton of probability is very
  useful in developing the theory of probability,
  it doesnt tell us how to compute probabilities
  of events.
• Classical Definition If S has a finite number of
  sample points and are equally likely to occur,
  then P(A) number of points in A / number in S.
• If S doesnt contain equally likely outcomes,
  then P(A) sum of the weights associated with
  points in A.

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Classical definition(contd)

• To use this definition, we need to calculate the
  number of points in S and in A.
• How do we do this without actually listing all
  possible outcomes?
• Using Counting Techniques.
• Principle of addition and multiplication
• Permutations and combinations

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Principle of Addition and Multiplication

• If the task is done if any one of the subtasks is
  done, then the total number of ways of doing the
  main task is n1 n2 ... nk .
• If the task is done if and only if all the
  subtasks are done, then the total number of ways
  of doing the main task is the product n1 ? n2 ?
  ... ? nk.

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PERMUTATION

• Suppose that r objects are to be drawn without
  replacement from n (r ? n).
• If the order of selection is important, then
  using the principle of multiplication we see that
  the total number of ways of doing this is
  n(n-1)...(n-r1).
• This could be written in a compact form using the
  factorials as n!/(n-r)! or Prn.

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COMBINATION

• Suppose that r objects are to be drawn without
  replacement from n (r ? n).
• When the order of selection is not important, any
  particular set of r objects can be ordered in Prr
  r! ways, the total number of ways of selecting
  r out of n in which order is immaterial is Prn
  /r!. It is convenient to denote this by Crn or by

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EXAMPLES
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CONDITIONAL PROBABILITY

• What we saw so far is referred to as
  unconditional probability. That is, the
  probabilities of events of interest were computed
  only based on the sample space and with no prior
  information.
• Sometimes it is convenient to compute certain
  unconditional probabilities by first conditioning
  on some event.

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CONDITIONAL PROBABILITY

• Also, this plays an important role in stochastic
  modeling.
• In a finite buffer queuing model, computation of
  waiting time of an admitted customer involves
  conditional probability.
• DEFINITION P(B/A) P(A?B) / P(A)
• Events A and B are independent if and only if
  P(A?B) P(A)P(B).

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RANDOM VARIABLES

• Often in probability and statistics, the
  quantities that are of interest are not the
  outcomes but rather the values associated with
  the outcome of the experiment.
• If n items are selected from a production lot the
  quality inspector is interested in the number of
  defectives out of the n chosen and the
  corresponding probabilities.

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RANDOM VARIABLES (contd)

• A random variable, X, is a real-valued function
  defined on the sample space S into the set of
  real numbers.
• Random variables can be
• Discrete taking only discrete values
• Number of defective molds
• Continuous taking continuous values
• Time taken to assemble a product
• Mixture of discrete and continuous
• Waiting time of a customer

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INDEPENDENCE
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EXAMPLES
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STUDY OF RANDOM VARIABLES

• Probability functions
• Probability mass function (discrete)
• Probability density function (continuous)
• Cumulative probability distribution function

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