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.