
Solutions 205
To simplify the notation, we will first transform the three variables to give
them a zero marginal expectation and a unity marginal variance.
e
G =
G − E(G)
p
VAR(G)
=
G − 50
10
e
E =
E − E(E)
p
VAR(E)
=
E − 50
10
e
V =
V − E(V)
p
VAR(V)
=
V − 50
10
The resulting normalised variables are
e
G ∼ N (0, 1) ,
e
E ∼ N (0, 1) ,
e
V |
e
G,
e
E ∼ N
1
2
e
G +
r
1
2
e
E,
1
2
2
!
.
We are now able to compute the joint density of the three transformed vari-
ables
f
e
G = g,
e
E = e,
e
V = v
∝ f
e
G = g
+ f
e
E = e
+ f
e
V = v |
e
G = g,
e
E = e
= −
g
2
2
−
e
2
2
− 2
v −
1
2
g −
r
1
2
e
!
2
= −
g
e
v
T
1
√
2
2
−1
√
2
2
3
2
−
√
2
−1 −
√
2 2
g
e
v
= −
1
2
g
e
v
T
1 0
1
2
0 1
q
1
2
1
2
q
1
2
1
−1
g
e
v
.
Now
VAR
e
G
e
E
e
V
=
1 0
1
2
0 1
q
1
2
1