Student Notes To Accompany MS4214: STATISTICAL INFERENCE

Now consider the random experiment which consists of picking 3 people at random
from the 1997 electoral register for Limerick. The outcome of such an experiment will
be a collection of 3 human beings and the set S consists of all subsets of 3 human
beings which may be formed from the set of all human beings whose names are on the
register. We can clearly envisage an infinite sequence of independent repetitions of such
an experiment. Consider the random vector X = (X1, X2, X3) where for i = 1, 2, 3,
Xi = 0 if the ith person chosen is a male and Xi = 1 if the ith person chosen is a
female. When we say that X1, X2, X3 are independent and identically distributed or IID
with P (Xi = 1) = θ we are taken to mean that in an infinite sequence of independent
repetitions of the experiment the proportion of outcomes which produce, for instance,
a value of X = (1, 1, 0) is given by θ 2 (1 − θ).
Suppose that the value of θ is unknown and we propose to estimate it by the
estimator ˆ θ whose value is given by the proportion of females in the sample of size 3.
Since ˆ θ depends on the value of X we sometimes write ˆ θ(X) to emphasise this fact.
We can work out the probability distribution of ˆ θ as follows :
X P (X = x) ˆ θ(x)
(0, 0, 0) (1 − θ) 3 0
(0, 0, 1) θ(1 − θ) 2 1/3
(0, 1, 0) θ(1 − θ) 2 1/3
(1, 0, 0) θ(1 − θ) 2 1/3
(0, 1, 1) θ 2 (1 − θ) 2/3
(1, 0, 1) θ 2 (1 − θ) 2/3
(1, 1, 0) θ 2 (1 − θ) 2/3
(1, 1, 1) θ 3 1
Thus P ( ˆ θ = 0) = (1 − θ) 3 , P ( ˆ θ = 1/3) = 3θ(1 − θ) 2 , P ( ˆ θ = 2/3) = 3θ 2 (1 − θ) and
P ( ˆ θ = 1) = θ 3 . We now ask whether ˆ θ is a good estimator of θ? Clearly if θ = 0 we
have that P ( ˆ θ = θ) = P ( ˆ θ = 0) = 1 which is good. Likewise if θ = 1 we also have that
P ( ˆ θ = θ) = P ( ˆ θ = 1) = 1. If θ = 1/3 then P ( ˆ θ = θ) = P ( ˆ θ = 1/3) = 3(1/3)(1−1/3) 2 =
4/9. Likewise if θ = 2/3 we have that P ( ˆ θ = θ) = P ( ˆ θ = 2/3) = 3(2/3) 2 (1−2/3) = 4/9.
However if the value of θ lies outside the set {0, 1/3, 2/3, 1} we have that P ( ˆ θ = θ) = 0.
Since ˆ θ is a random variable we might try to calculate its expected value E( ˆ θ) i.e.
the average value we would get if we carried out an infinite number of independent
repetitions of the experiment. We have that
E( ˆ θ) = 0P ( ˆ θ = 0) + (1/3)P ( ˆ θ = 1/3) + (2/3)P ( ˆ θ = 2/3) + 1P ( ˆ θ = 1) ,
= 0(1 − θ) 3 + (1/3)3θ(1 − θ) 2 + (2/3)3θ 2 (1 − θ) + 1θ 3 ,
= θ.
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