1
Probability Theory
1.1
Set Theory
Denition 1.1. Sample Space S : The set of all possible outcomes of a particular experiment.
Example 1.1. Tossing a coin: S =cfw_H, T,
Rolling a die: S =cfw_1, 2, 3, 4, 5, 6
Denition 1.2. Event: Any collection of possibl
5
Properties of a Random Sample
5.1
Basic Concepts of Random Samples
Denition 5.1. Random variables X1 , , Xn : are a random sample from the population f (x) if
X1 , , Xn are independent and identically distributed (iid) with pdf (or pmf ) f (x).
n
f ( x
6
Principles of Data Reduction
6.2
6.2.1
The Suciency Principle
Sucient Statistics
Denition 6.1. T (X ) is a sucient statistic (SS) for if the conditional distribution of X given
T = t is independent of .
If T (X ) is a SS for , and T (x) = T (y ) for x
7
7.1
Point Estimation
Introduction
Denition 7.1.
An estimator is a function of a sample. (W (X1 , , Xn )
An estimate is an observed value of the estimator. (W (x1 , , xn )
7.2
7.2.1
Methods of Finding Estimators
Method of Moments
X
EX
EX 2
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1
n
n
8
Hypothesis Testing
8.1
Introduction
A statistical hypothesis is a statement about a population parameter.
eg. A new improved laundry soap is claimed to have a better cleaning power than the old
formula. Examine whether or not the data substantiate the