Geology Reference
In-Depth Information
EXERCISES IN LINEAR GEOSTATISTICS
1. Probabilities
Question 1.1
The variance of a random variable
x
is, by definition,
V
(
x
) =
E
[
x
-
]
2
#
.
Show that
V
(
x
) =
E
(
x
2
) - {
E
(
x
)}
2
Answer
Va r (
X
) =
where
=
E
(
x
) =
xf
()
x dx
2
#
(
x
)
f
( )
x dx
2
2
#
#
#
xfxdx
()
2
xfxdx
()
fxdx
()
=
=
E
(
x
2
)
- 2 ' +'
2
.1
=
E
(
x
2
)
-
2
=
E
(
x
2
)
- {
E
(
x
)}
2
Question 1.2
Let
x
be a random variable and “
a
” a constant.
(a) Show that
E
(
ax
) =
a
E
(
x
).
(b) Show that Var(
ax
) =
a
2
Var(
x
).
Answer
(a)
E
(
ax
)=
# #
(b) Var(
ax
)=
E
[(
ax
)
2
] -
E
[(
ax
)]
2
=
a
2
E
(
x
2
) -
a
2
( ) ()
ax f
x dx
a
xf
()
x dx
a
aE x
()
2
=
a
2
V
(
x
)
Question 1.3
Let
x
,
y
and
z
be three random variables such that
z
=
x
+
y
.
Show that
V
(
z
) =
V
(
x
) +
V
(
y
) + 2Cov(
x
,
y
)
Answer
##
E
(
x
+
y
)=
(
xyfxy x y
)
( ,
)
.
##
##
xf
(, ) .
x y dx dy
yf
(, ) .
x y dx dy
=
##
##
x f
[(,) ]
x y dy dx
f
(,) ..
x y dx y dy
=
## #
E
[(
x
+
y
) =
E
(
x
) +
E
(
y
)
xf xdx
.()
yf y dy
.( .
=
V
(
x
+
y
) =
E
[(
x
+
y
)
2
] - [
E
(
x
+
y
)]
2
Also
