0Introduction

IB Statistics

0 Introduction

Statistics is a set of principles and procedures for gaining and processing quan-

titative evidence in order to help us make judgements and decisions. In this

course, we focus on formal statistical inference. In the process, we assume that

we have some data generated from some unknown probability model, and we aim

to use the data to learn about certain properties of the underlying probability

model.

In particular, we perform parametric inference. We assume that we have

a random variable

X

that follows a particular known family of distribution

(e.g. Poisson distribution). However, we do not know the parameters of the

distribution. We then attempt to estimate the parameter from the data given.

For example, we might know that

X ∼ Poisson

(

µ

) for some

µ

, and we want

to figure out what µ is.

Usually we repeat the experiment (or observation) many times. Hence we

will have

X

1

, X

2

, ··· , X

n

being iid with the same distribution as

X

. We call the

set X = (X

1

, X

2

, ··· , X

n

) a simple random sample. This is the data we have.

We will use the observed X = x to make inferences about the parameter

θ

,

such as

– giving an estimate

ˆ

θ(x) of the true value of θ.

– Giving an interval estimate (

ˆ

θ

1

(x),

ˆ

θ

2

(x)) for θ

– testing a hypothesis about θ, e.g. whether θ = 0.