A Brief Introduction to Mathematical Statistics

Xiaoheng YangTokyo Institute of Technology

Structure

Basic concepts Sample and The whole Histogram and Experience distribution

function Statistics quantities and distribution Next…

Basic ConceptsWhat’s Mathematical Statistics?

“Statistics” “Status”

Mathematical Statistics Statistics difference

Eg.1 (i) Record unqualified goods and compute the wastrel rate from all the products in one month; (ii) Select some sample goods randomly from all the ones then predict the wastrel rate in one month. Eg.2 (i) Investigate and make the sick list according to a specific disease from all the population in one area, then calculate the percentage of the sicken persons for this area ; (ii) Select n persons randomly and make the sick list according to a specific disease, then predict the percentage of the sicken persons for this area.

Basic ConceptsEg.1 (i) Record unqualified goods and compute the wastrel rate from all the products in one month;Eg.2 (i) Investigate and make the sick list according to a specific disease from all the population in one area, then calculate the percentage of the sicken persons for this area ;

Eg.1(ii) Select some sample goods randomly from all the ones then predict the wastrel rate in one month. Eg.2(ii) Select n persons randomly and make the sick list according to a specific disease, then predict the percentage of the sicken persons for this area.

Statistics

Mathematical Statistics

certainty

uncertainty : data, processing method, result

Basic Concepts

What’s Mathematical Statistics?

Object: stochastic data

Task: collecting, organizing, & analyzing data;

then predicting & deducing;

finally, making decision based on above.

Basic Concepts

Is Mathematical Statistics necessary or not?

Eg.1 (i) Record unqualified goods and compute the wastrel rate from all the products in one month; Eg.2 (i) Investigate and make the sick list according to a specific disease from all the population in one area, then calculate the percentage of the sicken persons for this area ;

energy , time cost and sometime impossible

Eg.1(ii) Select some sample goods randomly from all the ones then predict the wastrel rate in one month. Eg.2(ii) Select n persons randomly and make the sick list according to a specific disease, then predict the percentage of the sicken persons for this area.

Basic Concepts

Analyzing the randomicity for data

Sources: chanciness, Eg.2.ii; uncertainty.

Application areas: Industries, Agriculture, Medicals, Military, Economy…

Fundamental contents: Data gathering, including sampling theory, test decision...; Statistical prediction, including estimation & verification.

Validity of methods: Probability Mathematical Statistics

Sample and Whole/Collectivity

nXXXX ......,3,2,1

Samples: observed data n means sample capacity

Sample space: All possible samples consist of the sample space

Pay attention to the property of sample: Duplicity or twoness (specific data or random variables); Independent & same probability distribution.

Eg.3 Lottery problem

Whole/collectivity: The whole means a random variable.

Eg.4 Lathe problem

X

Sample and Whole/Collectivity

Predicting the probability distribution: experience determined distribution parameter decision

Eg.5 ……

Parameter space

parameter estimation; unparameter estimation.

Histogram & Experience Distribution Function

X nXXXX ......,3,2,1

max)(min)1( , XXXX n

Histogram:

Step.1 finding…

Step.2 dividing…

Step.3 recording…

Step.4 Calculating…

Histogram & Experience Distribution Function

X nXXXX ......,3,2,1

)()1()2()1( ... nn XXXX

Experience Distribution Function:

)(

)1()(

)1(

,1

1,...,2,1,,

,0

)(

n

kkn

Xx

nkXxXn

k

Xx

xF

Prove…

Statistical quantities & distribution

nXXXX ......,3,2,1 )()1()2()1( ... nn XXXX

Statistical quantities: unorganized data organized data processing

YX

n

iii

n

i

kik

n

i

kik

n

ii

n

ii

SS

S

YYXXn

S

XXn

B

Xn

A

SS

XXn

S

Xn

X

12

112

1

1

2

2

1

2

1

))((1

)(1

1

)(1

1

Statistical quantities & distribution

ondistributi2nXXXX ......,3,2,1

n

iiXY

1

2

are independent and obey the normal distribution N(0,1)

Theory.1: are independent and obey the normal distribution N(0,1), , j 1,2,…k, then

kYYY ,..., 21

)(~ 2jnY

)(~1

2

1

k

jj

k

jj nYY

Theory.2:

0,0

0,

)2

(2

1

)()(

2)(,)()(

)2/(12/

2

x

xexn

xpii

nXDnXEi

xnn

Statistical quantities & distribution

)(~),1,0(~ 2 nYNX

nY

XT

/

t (student) distribution: ,X and Y are independent, then t distribution is

xxnn

npdf n ,)2/1(

)2/(

)2/)1( 2/)1(2

Statistical quantities & distribution

0,0

0,)()2/()2/(

)2/)(2/)(

21

12/

222121

21

21

121

x

xnxn

xnn

nn

nnpdf nn

nnn

)(~),(~ 22

12 nYnX

2

1

/

/

nY

nXF

F distribution: ,X and Y are independent, then F distribution is

Normal distribution

),(~ 2uNX

tindependenSX

nnS

nuNX

2

222

2

,)3(

)1(~/)2(

)/,(~)1(

Theory:

thennXXXX ......,3,2,1

Prove….

Next…