MATH408: Probability

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MATH408 Probability StatisticsSummer
1999WEEK 5
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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Joint PDF

• So far we saw one random variable at a time.
  However, in practice, we often see situations
  where more than one variable at a time need to be
  studied.
• For example, tensile strength (X) and diameter(Y)
  of a beam are of interest.
• Diameter (X) and thickness(Y) of an
  injection-molded disk are of interest.

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Joint PDF (Contd)X and Y are continuous

• f(x,y) dx dy P( x lt X lt xdx, y lt Y lt ydy) is
  the probability that the random variables X will
  take values in (x, xdx) and Y will take values
  in (y,ydy).
• f(x,y) ? 0 for all x and y and

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Measures of Joint PDF
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Independence
We say that two random variables X and Y are
independent if and only if P(X?A, Y?B)
P(X?A)P(Y?B) for all A and B.
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EXAMPLES
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Groundwork for Inferential Statistics

• Recall that, our primary concern is to make
  inference about the population under study.
• Since we cannot study the entire population we
  rely on a subset of the population, called
  sample, to make inference.
• We saw how to take samples.
• Having taken the sample, how do we make inference
  on the population?

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Basic Concepts
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Figure 3-36 (a) Probability density function of a
pull-off force measurement in Example 3-33.
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Figure 3-36 (b) Probability density function of
the average of 8 pull-off force measurements
in Example 3-33.
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Figure 3-36 (c) Probability density Probability
density function function of the sample
variance of 8 pull-off force measurements in
Example 3-33.
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An important result
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Examples
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Central Limit Theorem

• One of the most celebrated results in Probability
  and Statistics
• History of CLT is fascinating and should read
  The Life and Times of the Central Limit Theorem
  by William J. Adams
• Has found applications in many areas of science
  and engineering.

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CLT (contd)

• A great many random phenomena that arise in
  physical situations result from the combined
  actions of many individual ones.
• Shot noise from electrons holes in a vacuum tube
  or transistor atmospheric noise, turbulence in a
  medium, thermal agitation of electrons in a
  conductor, ocean waves, fluctuations in stock
  market, etc.

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CLT (contd)

• Historically, the CLT was born out of the
  investigations of the theory of errors involved
  in measurements, mainly in astronomy.
• Abraham de Moivre (1667-1754) obtained the first
  version.
• Gauss, in the context of fitting curves,
  developed the method of Least Squares, which lead
  to normal distribution.

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Examples
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HOMEWORK PROBLEMS

• Sections 3.11 through 3.12
• 109,111, 114-116-119, 121-123, 129-130

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Examples
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Tests of Hypotheses

• Two types of hypotheses Null (H0)and alternative
  (H1)

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Basic Ideas in Tests of Hypotheses

• Set up H0 and H1. For a one-sided case, make sure
  these are set correctly. Usually these are done
  such that type 1 error becomes costly error.
• Choose appropriate test statistic. This is
  usually based on the UMV estimator of the
  parameter under study.
• Set up the decision rule if ? P(type 1 error)
  is specified. If not, report a p-value.
• Choose a random sample and make the decision.

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Setting up Ho and H1

• Suppose that the manufacturer of airbags for
  automobiles claims that the mean time to inflate
  airbag is no more than 0.1 second.
• Suppose that the costly error is to conclude
  erroneously that the mean time is lt 0.1.
• How do we set up the hypotheses?

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ILLUSTRATIVE EXAMPLE
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Test on µ using normal

• Sample size is large
• Sample size is small, population is approximately
  normal with known ?.

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DNR Region
µ
CP_1
CP_2
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Computation of P(type 2 error)
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Example (page 142)

• µ Mean propellant burning rate (in cm/s).
• H0µ 50 vs H1µ ? 50.
• Two-sided hypotheses.
• A sample of n10 observations is used to test the
  hypotheses.
• Suppose that we are given the decision rule.
• Question 1 Compute P(type 1 error)
• Question 2 Compute P(type 2 error when µ 52.

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DECISION RULE
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Calculation of P(type 1 error)
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Example
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Confidence Interval

• Recall point estimate for the parameter under
  study.
• For example, suppose that µ mean tensile
  strength of a piece of wire.
• If a random sample of size 36 yielded a mean of
  242.4psi.
• Can we attach any confidence to this value?
• Answer No! What do we do?

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Confidence Interval (contd)

• Given a parameter, say, , let denote its
  UMV estimator.
• Given ?, 100(1- ? ) CI for is constructed
  using the sampling (probability) distribution of
  as follows.
• Find L and U such that P(L lt lt U) 1- ?.
• Note that L and U are functions of .