Exercises
Chapter 2
2.1 Marginal and conditional probability: The social mobility data from Sec
tion 2.5 gives a joint probability distribution on (
Y
1
, Y
2
)= (father’s oc
cupation, son’s occupation). Using this joint distribution, calculate the
following distributions:
a) the marginal probability distribution of a father’s occupation;
b) the marginal probability distribution of a son’s occupation;
c) the conditional distribution of a son’s occupation, given that the father
is a farmer;
d) the conditional distribution of a father’s occupation, given that the
son is a farmer.
2.2 Expectations and variances: Let
Y
1
and
Y
2
be two independent random
variables, such that E[
Y
i
] =
μ
i
and Var[
Y
i
] =
σ
2
i
. Using the definition of
expectation and variance, compute the following quantities, where
a
1
and
a
2
are given constants:
a) E[
a
1
Y
1
+
a
2
Y
2
] ,
Var[
a
1
Y
1
+
a
2
Y
2
];
b) E[
a
1
Y
1

a
2
Y
2
] ,
Var[
a
1
Y
1

a
2
Y
2
].
2.3 Full conditionals: Let
X, Y, Z
be random variables with joint density (dis
crete or continuous)
p
(
x, y, z
)
/
f
(
x, z
)
g
(
y, z
)
h
(
z
). Show that
a)
p
(
x

y, z
)
/
f
(
x, z
), i.e.
p
(
x

y, z
) is a function of
x
and
z
;
b)
p
(
y

x, z
)
/
g
(
y, z
), i.e.
p
(
y

x, z
) is a function of
y
and
z
;
c)
X
and
Y
are conditionally independent given
Z
.
2.4 Symbolic manipulation: Prove the following form of Bayes’ rule:
Pr(
H
j

E
) =
Pr(
E

H
j
) Pr(
H
j
)
P
K
k
=1
Pr(
E

H
k
) Pr(
H
k
)
where
E
is any event and
{
H
1
, . . . , H
K
}
form a partition. Prove this using
only axioms
P1

P3
from this chapter, by following steps a)d) below:
a) Show that Pr(
H
j

E
) Pr(
E
) = Pr(
E

H
j
) Pr(
H
j
).
P.D. Ho
↵
,
A First Course in Bayesian Statistical Methods
,
Springer Texts in Statistics, DOI 10.1007/9780387924076
BM2,
c Springer Science+Business Media, LLC 2009
226
Exercises
b) Show that Pr(
E
) = Pr(
E
\
H
1
) + Pr(
E
\
{
[
K
k
=2
H
k
}
).
c) Show that Pr(
E
) =
P
K
k
=1
Pr(
E
\
H
k
).
d) Put it all together to show Bayes’ rule, as described above.
2.5 Urns: Suppose urn
H
is filled with 40% green balls and 60% red balls, and
urn
T
is filled with 60% green balls and 40% red balls. Someone will flip
a coin and then select a ball from urn
H
or urn
T
depending on whether
the coin lands heads or tails, respectively. Let
X
be 1 or 0 if the coin lands
heads or tails, and let
Y
be 1 or 0 if the ball is green or red.
a) Write out the joint distribution of
X
and
Y
in a table.
b) Find E[
Y
]. What is the probability that the ball is green?
c) Find Var[
Y

X
= 0], Var[
Y

X
= 1] and Var[
Y
]. Thinking of variance as
measuring uncertainty, explain intuitively why one of these variances
is larger than the others.
d) Suppose you see that the ball is green. What is the probability that
the coin turned up tails?
2.6 Conditional independence: Suppose events
A
and
B
are conditionally in
dependent given
C
, which is written
A
?
B

C
. Show that this implies that
A
c
?
B

C
,
A
?
B
c

C
, and
A
c
?
B
c

C
, where
A
c
means “not
A
.” Find an
example where
A
?
B

C
holds but
A
?
B

C
c
does not hold.
2.7 Coherence of bets: de Finetti thought of subjective probability as follows:
Your probability
p
(
E
) for event
E
is the amount you would be willing to
pay or charge in exchange for a dollar on the occurrence of
E
. In other
words, you must be willing to
•
give
p
(
E
) to someone, provided they give you $1 if
E
occurs;
•
take
p
(
E