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S TATISTIKA UGM Y OGYAKARTA I ntroduction 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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T HE U SE OF M ATHEMATICAL S TATISTICS

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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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S TATISTICAL M ETHODS 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.

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I NTRODUCTION OF R ELIABILITY 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 > t ) = 1 – F x (t) Lee J. Bain & Max Engelhardt

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M AXIMUM L IKELIHOOD E STIMATION IMS is introduces us to the MLE, Let L(0) = f (x 1,....,x n :0), 0 Є, be the joint pdf of X 1,....,X n. For a given set bof observatios, (x 1,....,x n :0), a value in Ω at which L (0) is a maximum and called the maximum likelihood estimate of θ. That is, is a value of 0 that statifies f (x 1,....,x n : ) = max f (x 1,....,x n :0), Lee J. Bain & Max Engelhardt

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A NALYSIS OF L IFE -T ESTING M ODELS 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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N ONPARAMETRIC S TATISTICAL 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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E XAMPLE 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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A PPROXIMATION FOR T HE B INOMIAL D ISTRIBUTION 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 P[X 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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A SYMPTOTIC N ORMAL D ISTRIBUTIONS If Y 1, Y 2, … is a sequence of random variables and m and c are constants such that as, then Y n is said to have an asymptotic normal distribution with asymptotic mean m and asymptotic variance c 2 /n. Example: The random sample involve n = 40 lifetimes of electrical parts, X i ~ EXP(100). By the CLT, has an asymptotic normal distribution with mean m = 100 and variance c 2 /n = / 40 = 250.

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A SYMPTOTIC D ISTRIBUTION OF C ENTRAL O RDER S TATISTICS Theorem Let X 1, …, X n be a random sample from a continuous distribution with a pdf f(x) that is continuous and nonzero at the pth percentile, x p, for 0 < p < 1. If k/n p (with k – np bounded), then the sequence of kth order statistics, X k:n, is asymptotically normal with mean x p and variance c 2 /n, where Example Let X 1, …, X n be a random sample from an exponential distribution, X i ~ EXP(1), so that f(x) = e -x and F(x) = 1 – e -x ; x > 0. For odd n, let k = (n+1)/2, so that Y k = X k:n is the sample median. If p = 0.5, then the median is x 0.5 = - ln (0.5) = ln 2 and Thus, X k:n is asymptotically normal with asymptotic mean x 0.5 = ln 2 and asymptotic variance c 2 /n = 1/n.

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T HEOREM If then Proof

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T HEOREM For a sequence of random variables, if then For the special case For the special case Y = c, the limiting distribution is the degenerate distribution P[Y = c] = 1. this was the condition we initially used to define stochastic convergence., then for any function g(y) that is continuous at c, If

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T HEOREM If X n and Y n are two sequences of random variables such thatand then: Example Suppose that Y~BIN(n, p). Thus it follows that

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Theorem Slutskys Theorem If X n and Y n are two sequences of random variables such that and Note that as a special case X n could be an ordinary numerical sequence such as X n = 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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