Statistika UGM Yogyakarta

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Statistika UGM Yogyakarta
TUGAS
TUGAS

• Introduction of Mathematical Statistics 2

By Indri Rivani Purwanti (10990) Gempur Safar
(10877) Windu Pramana Putra Barus
(10835) Adhiarsa Rakhman (11063)
Dosen Prof.Dr. Sri Haryatmi Kartiko, S.Si.,
M.Sc.
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The Use ofMathematical Statistics
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• Introduction to Mathematical Statistics (IMS) can
  be applied for the whole statistics subject,
  such as
• Statistical Methods I and II
• Introduction to Probability Models
• Maximum Likelihood Estimation
• Waiting Times Theory
• Analysis of Life-testing models
• Introduction to Reliability
• Nonparametric Statistical Methods
• etc.

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Statistical Methods

• In Statistical Methods, Introduction of
  Mathematical Statistics are used to
• introduce and explain about the random variables
  , probability models and the suitable cases which
  can be solve by the right probability models.
• How to determine mean (expected value), variance
  and covariance of some random variables,
• Determining the convidence intervals of certain
  random variables
• Etc.

Lee J. Bain Max Engelhardt
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Probability Models

• Mathematical Statistics also describing the
  probability model that being discussed by the
  staticians.
• The IMS being used to make student easy in
  mastering how to decide the right probability
  models for certain random variables.

Lee J. Bain Max Engelhardt
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Introduction of Reliability

• The most basic is the reliability function that
  corresponds to probability of failure after time
  t.
• The reliability concepts
• If a random variable X represents the lifetime of
  failure of a unit, then the reliability of the
  unit t is defined to be
• R (t) P ( X gt t ) 1 F x (t)

Lee J. Bain Max Engelhardt
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Maximum Likelihood Estimation

• IMS is introduces us to the MLE,
• Let L(0) f (x1,....,xn0), 0 ? ?, be the joint
  pdf of X1,....,Xn. For a given set bof
  observatios, (x1,....,xn0), a value in O at
  which L (0) is a maximum and called the maximum
  likelihood estimate of ?. That is , is a
  value of 0 that statifies
• f (x1,....,xn )
  max f (x1,....,xn0),

Lee J. Bain Max Engelhardt
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Analysis of Life-Testing Models

• Most of the statistical analysis for parametric
  life-testing models have been developed for the
  exponential and weibull models.
• The exponential model is generally easier to
  analyze because of the simplicity of the
  functional form.
• Weibull model is more flexibel , and thus it
  provides a more realistic model in many
  applications , particularly those involving
  wearout and aging.

Lee J. Bain Max Engelhardt
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Nonparametric Statistical methods

• The IMS also introduce to us the nonparametrical
  methods of solving a statistical problem, such
  as
• one-sample sign test
• Binomial Test
• Two-sample sign test
• wilcoxon paired-sample signed-rank test
• wilcoxon and mann-whitney tests
• correlation tests-tests of independence
• wald-wolfowitz runs test
• etc.

Lee J. Bain Max Engelhardt
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• KETERKAITAN
• KONVERGENSI

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Example

• We consider the sequence of standardized
  variables

With the simplified notation
By using the series expansion
Where d(n) ? 0 as n ?
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Approximation for The Binomial Distribution
Example A certain type of weapon has probability
p of working successfully. We test n weapons, and
the stockpile is replaced if the number of
failures, X, is at least one. How large must n be
to have PX 1 0.99 when p 0.95?Use normal
approximation.
X number of failures p probability of working
successfully 0.95 q probability of working
failure 0.05
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Asymptotic Normal Distributions
If Y1, Y2, is a sequence of random variables
and m and c are constants such that
as
, then Yn is said to have an asymptotic normal
distribution with asymptotic
mean m and asymptotic variance c2/n.
Example The random sample involve n 40
lifetimes of electrical parts, Xi EXP(100). By
the CLT,
has an asymptotic normal distribution with mean m
100 and variance c2/n 1002/ 40 250.
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Asymptotic Distribution of Central Order
Statistics

• Theorem
• Let X1, , Xn be a random sample from a
  continuous distribution with a pdf f(x) that is
  continuous and nonzero at the pth percentile, xp,
  for 0 lt p lt 1. If k/n ? p (with k np bounded),
  then the sequence of kth order statistics, Xkn,
  is asymptotically normal with mean xp and
  variance c2/n, where

• Example
• Let X1, , Xn be a random sample from an
  exponential distribution, Xi EXP(1), so that
  f(x) e-x and F(x) 1 e-x x gt 0. For odd n,
  let k (n1)/2, so that Yk Xkn is the sample
  median. If p 0.5, then the median is x0.5 -
  ln (0.5) ln 2 and

Thus, Xkn is asymptotically normal with
asymptotic mean x0.5 ln 2 and asymptotic
variance c2/n 1/n.
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Theorem

• If

then
Proof
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Theorem

• If

, then for any function g(y) that is continuous
at c,
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Theorem

• If Xn and Yn are two sequences of random
  variables such that

and
then
Example Suppose that YBIN(n, p).
Thus it follows that
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Theorem

Slutskys Theorem If Xn and Yn are two sequences
of random variables such that
and
Note that as a special case Xn could be an
ordinary numerical sequence such as Xn n/(n-1).

then for any continuous function g(y),

and if g(y) has a nonzero derivative at y m,
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