Student Notes To Accompany MS4214: STATISTICAL INFERENCE

Example 1.2 (Tuberculosis). Suppose we are going to examine n people and record
a value 1 for people who have been exposed to the tuberculosis virus and a value 0
for people who have not been so exposed. The data will consist of a random vector
X = (X1, X2, . . . , Xn) where Xi = 1 if the ith person has been exposed to the TB virus
and Xi = 0 otherwise.
A Bernoulli random variable X has probability mass function
P (X = x|θ) = θ x (1 − θ) 1−x , (1.2)
for x = 0, 1 and θ ∈ (0, 1). A possible model would be to assume that X1, X2, . . . , Xn
behave like n independent Bernoulli random variables each of which has the same
(unknown) probability θ of taking the value 1.
Let x = (x1, x2, . . . , xn) be a particular vector of zeros and ones. Then the model
implies that the probability that the random vector X takes the value x is given by
P (X = x|θ) =
n�
θ xi 1−xi (1 − θ)
i=1
= θ �n i=1 xi n−
(1 − θ) �n i=1 xi .
Once again our plan is to use the value x of X that we actually observe to learn
something about the value of θ. �
Example 1.3 (Viagra). A chemical compound Y is used in the manufacture of Viagra.
Suppose that we are going to measure the micrograms of Y in a sample of n pills. The
data will consist of a random vector X = (X1, X2, . . . , Xn) where Xi is the chemical
content of Y for the ith pill.
A possible model would be to assume that X1, X2, . . . , Xn behave like n independent
random variables each having a N (µ, σ 2 ) density with unknown mean parameter µ ∈ R,
(really, here µ ∈ R + ) and known variance parameter σ 2 < ∞. Each Xi has density
fXi (xi|µ) =
1
√ 2πσ 2 exp
�
− (xi − µ) 2
2σ 2
Let x = (x1, x2, . . . , xn) be a particular vector of real numbers. Then the model implies
the joint density
fX(x|µ) =
=
n�
i=1
�
.
1
√
2πσ2 exp
�
− (xi − µ) 2 �
1
( √ 2πσ2 exp
) n
�
−
2σ 2
� n
i=1 (xi − µ) 2
2σ 2
Once again our plan is to use the value x of X that we actually observe to learn
something about the value of µ. �
5
�