S TATIS TIK A UGM YOGYAK AR TA

Introduction of Mathematical

S tatis tics 2

By :

Indri R ivani Purwanti (10990) Gempur Safar (10877)

Windu Pramana Putra Barus (10835) Adhiarsa Rakhman (11063)

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THE US E OF

Introduction to M athematical Statistics (I M S) can be applied for the whole statistics subject, such as:

 Statistical M ethods I and II

 Introduction to Probability M odels

 M aximum L ikelihood E stimation

 Waiting Times Theory

 Analysis of L ife-testing models

 Introduction to R eliability

Nonparametric Statistical M ethods

S TATIS TIC AL METHODS

In Statistical M ethods, I ntroduction of M athematical 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.

 _{H ow to determine mean (expected value), variance and}

covariance of some random variables,

 _{Determining the convidence intervals of certain random}

variables

 E tc.

Probability M odels

M athematical Statistics also describing the probability model that being discussed by the staticians.

INTR ODUC TION OF R E LIAB ILITY

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)

MAXIMUM LIK E LIHOOD

E S TIMATION

IM S is introduces us to the M L E ,

L et L (0) = f (x₁,....,x_n:0), 0 Є Ω, be the joint pdf of X₁,....,X_n. For a given set bof observatios, (x₁,....,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

ANALYS IS OF LIFE -TE S TING

MODE LS

M ost 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.

NONPAR AME TR IC S TATIS TIC AL

ME THODS

The IM S 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

E XAM P LE

We consider the sequence of ”standardized” variables:

( )

( n ) n n

With the simplified notation

σ =

_n

npq

By using the series expansion

AP P R OX IM ATION FOR THE B INO M IAL

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.

4 0.308 0.308 4 0.0025 0.25

122 ( )

ASYMPTOTIC NORMAL

as , then Y_n 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, X_i ~ EXP(100). By the CLT,

has an asymptotic normal distribution with mean m = 100 and variance c2_{/n =}

AS YM PTOTIC DIS TR IB UTION OF

asymptotically normal with mean x_p and variance c2_{/n, where}

2 asymptotic variance c2_{/n = 1/n.}

THEOR EM

p n

Y  → Y

For a sequence of random variables, if

then

d n

Y  → Y

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.

p n

Y  → c

( ) p ( )

n

g Y  → g c

Theorem

Slutsky’s Theorem If X_n and Y_n are two sequences of random variables such that