Foundations_of_Statistics-Rigby-Stasinopoulos.pdf

SUMMARY

MA2010 Foundations of Statistics

1. Probability

1 – 18

2. Number of ways to sample r items from n

19 - 26

3. Continuous Population Distributions

27 - 31

4. Exponential Population Distribution

32 – 44

5. Normal Population Distribution

45 – 55

6. Chi-square and t Distributions

56 – 61

7. Discrete Population Distributions

62 – 63

8. Poisson Population Distribution

64 – 72

9. Binomial Population Distribution

73 – 83

10. Geometric Population Distribution

84 – 89

11. Expectation and Variance of a random variable

90 – 101

12. Maximum Likelihood Estimation

102 – 120

13. The Simple Linear Regression Model: Theory

121 – 138

14. Multiple Linear Regression: Theory

139 – 154

15. Regression Models – Examples

155 – 177

16. Bibliography

178

Chapter 1

PROBABILITY

1.1 Set Theory Definitions

A SET is a collection of different elements e.g. A=set of scores on a dice={1,2,3,4,5,6}.

The SIZE n(A) of a set A is the number of elements it contains e.g. n(A) = 6.

An EXPERIMENT is a controlled process by which observations are obtained. Some

experiments that can be ‘independently and identically repeated’, e.g. tossing a dice

The set of all possible outcomes of an experiment is called the SAMPLE SPACE S

e.g. EXPERIMENT SAMPLE SPACE S

i) toss a dice and record the score {1,2,3,4,5,6}

ii) toss two coins and record the sequence of results {HH, HT, TH, TT}

iii) toss three coins and record the number of heads {0,1,2,3}

A subset A of the sample space S is called an EVENT .

A subset A with a SINGLE event is called a SIMPLE EVENT

An EVENT is said to OCCUR if any of the elements in A occurs e.g. toss a dice and record

the score S=(1,2,3,4,5,6) and let A=event of an even score=(2,4,6), then A is said to have

occurred if the result of tossing the dice is 2 or 4 or 6

SET NOTATION

A , the COMPLEMENT of A contains all elements of S not in A e.g. S = {1,2,3,4,5,6},

A = {2,4,6} then A = {1, 3, 5)

R. A. Rigby and D. M. Stasinopoulos  September 2005

A  B, the UNION of two sets A and B contains all elements that lie in A or B or both,

A  B is the event ‘A or B occurs’, e.g. A = {2, 4, 6} and B ={5, 6} then A  B = {2, 4, 5, 6}.

A  B

A  B, the INTERSECTION A and B contains all elements inside both A and B, A  B is the

event ‘A and B both occur’ e.g. A = {2, 4, 6} and B = {5, 6} then A  B = {6}.

A  B

Two sets A and B are DISJOINT (or MUTUALLY EXCLUSIVE ) if they have no elements

in common, i.e. if their intersection is the empty set  , i.e. if A  B =  .

Exercise : Draw a diagram for a sample space S with two events A and B and shade each of

the following events in a different colour:



1.2 Probability defined to be the limit of relative frequency

Let S be the sample space for an experiment and let A be an event in S. Suppose the

experiment is ‘independently and identically repeated’ n times. Let A

f count the frequency of

times event A occurs out of the n repetitions.

The PROBABILITY of event A occurring (in a single repetition of the experiment) is defined

to be the limiting value of the sample proportion A

of times event A occurs, i.e.

R. A. Rigby and D. M. Stasinopoulos  September 2005

p(A)



limit



limit









e.g. toss a fair coin n times and count H

f the frequency of times event H ‘heads’ occurs, then

p(H)



limit



limit











1.3 Axioms of probability

Let S be a sample space and A and B be two events in S. Probability satisfies the following

axioms:

A1 0<p(A)<1

probability lies between 0 and 1

A2 p(S)=1

S is certain to occur

A3 p(AuB)=p(A)+p(B) provided A and B are disjoint events

This is the Addition Rule for disjoint events

1.4 Rules derived from the axioms of probability

Many rules can be derived from the three basic axioms of probability, e.g.

Rule 1

p(  )=0

Rule 2

p(A ) = 1 - p(A)

Rule 3

p(A  B)

= p(A) + p(B) - p(A  B)

p(A or B) = p(A) + p(B) - p(A and B)

This is the General Addition Rule for any two events

Proof 3 p(A  B) = p(A) + p(A  B) since A and A  B are disjoint

p(B) = p(A  B) + p( A  B) since A  B and A  B are disjoint

R. A. Rigby and D. M. Stasinopoulos  September 2005

p(A  B) = p(A) + p(B) - p(A  B)

A 

1.5 Calculating probabilities from simple events

Rule 4: Let

A 

......,

n(A)



comprising n(A) simple events be an event in sample

space S

   



Examples:

i) Toss a fair dice and record the score S = {1, 2, 3, 4, 5, 6}

Let A = event ‘getting an odd score’ = {1, 3, 5}

p(A) = p(1) + p(3) + p(5) =





ii) Toss a fair coin twice and count the number of heads S = {0, 1, 2}

Let A = event ‘getting at least one head’ = {1, 2}

p(A) = p(1) + p(2) =





P(HH)=1/4

P(HT)=1/4

P(TH)=1/4

P(TT)=1/4

R. A. Rigby and D. M. Stasinopoulos  September 2005



n(A)

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