Lecture 1

Introduction

What is ‘Statistics’?

Statistics is the study of the collection, analysis, and interpretation

of data.

What is ‘Data’?

Data (plural of “datum”) are the results of measurements, can be

the basis of graphs, images, or observations of a set of variables.

What does the theory of Statistics do?

It provides a basis for the whole range of techniques that are used

within applications of Statistics.

Let’s get started!

Population is data that we are interested in.

Sample is a subset of a population.

Objective of Statistics is to make an inference about a population

based on information contained in a sample.

Graphical Approach

Example: We roll two dice and observe the sum of numbers.

Sum Combinations Number Probability

Numerical Methods

Numerical descriptive measures:

1. Measure of central tendency

2. Measure of dispersion or variation

Definition: The mean of a sample of n measured responses

is given by

The corresponding population mean is denoted by .

Example:

Definition: The variance of a sample of measurements

is given by

The corresponding population variance is denoted by .

The standard deviation of a sample is given by

Empirical Rule: For a distribution of measurements that is

approximately Normal, the interval with endpoints

Example:

Probability

Example: We roll a die.

Set Notation

We’ll use letter A, B, C,… to denote sets.

Let S denote the set of all elements under consideration. Then S is

the universal set.

We say A is a subset of B if every element in A is also in B.

The null, or empty set ( ) is the set consisting of no points. So

is a subset of any other set.

Venn Diagrams:

The union of A and B is the set of all points in A or in B.

The intersection of A and B is the set of all points in both A and B.

If , then the compliment of A is the set of all points in S that

are not in A.

Two set A and B are called disjoint or mutually exclusive, if they

do not have any elements in common.

Example:

The Distributive Laws:

De Morgan’s Laws:

Probabilistic Model

Consider a discrete case.

Definition: An experiment is the process by which an observation

is made.

The outcomes of an experiment are called events.

Example: We toss a die.

The event that can be decomposed into other events is called a

compound event. The event that cannot be decomposed is called a

simple event.

A simple event consists of only one point, called a sample point.

Definition: The sample space associated with an experiment is the

set consisting of all possible sample points, and denoted by S.

A discrete sample space contains wither a finite or a countable

number of distinct sample points.

Definition: Suppose S is a sample space associated with an

experiment. To every event A in S ( ), we assign a number,

P(A), called the probability of A, so that the following holds:

Proposition: ( )

Proof:

Problem: Use the proposition to prove that

( ) ∑ ( ) , .

Example: (#2.18) We toss two balanced coins.

Example: (#2.22) A, B are events, and . Prove that

( ) ( ) ( ̅)

Solution: