a brief introduction to mathematical statistics
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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…
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