Elements of Statistics - Lecture Notes | STAT 322, Exams of Statistics

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ELEMENTS OF STATISTICS

OUTLINE

Introduction

Sample Mean

Sample Variance

Questions and Solutions

Reading:

G. R. Cooper & C. D. McGillem 4.1 - 4.

EE/STAT 322, #

INTRODUCTION

Two branches of Statistics:

that can be easily understand;Descriptive statistics: collecting, grouping and presenting data in a way

parameters from the given data.Inductive statistics (or statistical inference): draw conclusion or estimate

EE/STAT 322, #

SAMPLE MEAN

Population:

the collection of data being studied. Size of population:

N

Sample:

the items begin selected for test. Size of sample:

n

.

Assume

X

i

are RVs from the population and each is assumed to have a

PDF

f

X

x

) .

Sample Mean:

X

n 1

n

i ∑

X

i ,

We use small

x

i

to denote a certain value of

X

i , so

x

n 1

i n

x

i .

EE/STAT 322, #

4

SAMPLE MEAN (CONT.)

Sample mean is

unbiased

estimate of the mean of the population.

Proof:

Let the true mean be denoted by

X

E

[

X

] =

E

[

n 1

n

i ∑

X

i ]

n 1

n

i ∑

E

[

X

i ] =

n 1

n

i ∑

X

X

Mean of the sample mean is the true mean.

EE/STAT 322, #

VARIANCE OF SAMPLE MEAN (CONT.)

1. Second, if we have a limited

N

(^) , and we sample without replacement.

var

X

σ

2

n

N

n

N

As

N

increases, we obtain the result for the first case.

As

n

increases, the variance of sample mean decreases

proportionally

EE/STAT 322, #

EXAMPLES AND SOLUTIONS

Example:

Samples of a waveform

are

RVs

X i = X ( t i ) ,

for

i

,... , n

X

and

σ

2

N

mean.that is only one percent of the truewith a standard deviation (STD)be taken to obtain a sample meanFind out how many samples should

t

)

( t

X 1

t

2

t

3

t

n

t

n

x

Solution:

var

X

σ

2

n

, and

σ

2

n

X

n^9

, and

n

EE/STAT 322, #

EXAMPLES AND SOLUTIONS (CONT.)

Example:

(Ex 4-2.1) In a production line with

N

, every 100th diode

is tested for current

I

−

1

and

I

1 .

(a) If

I

−

1

has a mean of

−

6

and variance

−

12

, how many diodes must

be tested to obtain a sample mean whose STD is 5% of the true mean?

(b)

E

[

I

1 ] = 0

and

σ

I 2 1

Find

n

such that the STD of sample

mean is 2% of the true mean?

(c) If the larger number

n

of (a) and (b) is used, find the STDs for both

tests.

EE/STAT 322, #

EXAMPLES AND SOLUTIONS (CONT.)

(a) Solution:

E

[

I

−

1 ] = 10

−

6 , and

σ

I 2 −

1

−

12

. The conditions tell us

var

Iˆ

−

1 }

σ

I 2 −

1

n

, and

σ

I 2 −

1

n

· E [ I − 1

] = 0

−

6

n

(c) ...(b) ... EE/STAT 322, #

11

EXAMPLES AND SOLUTIONS (CONT.)

(c) ...^ (b) ... EE/STAT 322, #

SAMPLE VARIANCE

The variance of the samples (

S

2 ) is another important performance metric

besides the variance of sample mean (

σ

2 ˆ

X

Definition:

S

2

n 1

i n

X

i

−

X

2

n 1

i n

[

X

i

−

n 1

i n

X

j ] 2 .

Mean of sample variance:

E [ S 2 ].

When

N

is not large.

E

[

S

2 ] =

N

N

(^) −

1

n

−

1

n

σ

2 .

By redefining

S

2

N

(^) −

1

N

n

n

−

1

S

2 , the bias can be removed.

EE/STAT 322, #

VARIANCE OF SAMPLE VARIANCE

Assume

N

is large. Variance of sample variance:

var

{ S 2 } = μ 4 − σ 4

n

where

μ

4

is the fourth central moment given by

μ

4

E

[(

X − X ) 4 ].

For Gaussian RVs,

μ

4

σ

4 .

Variance of

S

2 :

var

S 2 } = ( n

n − 1 ) 2 ·

var

{ S 2 } = n ( μ 4 − σ 4 )

n

2

EE/STAT 322, #

SAMPLE VARIANCE (CONT.)

Example:

Given

X

σ

2

, and the sample size

n

X

i }

are

Gaussian RVs. Find the variance of the unbiased sample variance

S

2 .

Solution:

var

S 2 } = n ( μ 4 − σ 4 )

( n

−

2

μ

4

σ

4

2 .

So var

S

2 }

900

· (

· 9 2 − 9 2 )

899

2

Observation:

the STD of

S

2

is

, which is

of the true variance.

In comparison, the STD of sample mean

X

is

of the true mean for

n

Measurement of sample variance is not as accurate as the sample mean,

given the same sample size

n

.

EE/STAT 322, #

SAMPLE VARIANCE (CONT.)

n

(

σ 4 − σ 4 )

( n

−

2

n

(

· 9 2 )

( n

−

2

n

, so

n

(b) ... EE/STAT 322, #