University of California, Los Angeles
Department of Statistics
Statistics 100A
Instructor: Nicolas Christou
Homework 7
EXERCISE 1
Show that
V ar
(
X

Y
) =
V ar
(
X
) +
V ar
(
Y
)

2
Cov
(
X,Y
).
EXERCISE 2
If
X
and
Y
are independent variables with equal variances ﬁnd
Cov
(
X
+
Y,X

Y
).
EXERCISE 3
If
U
=
a
+
bX
and
V
=
c
+
dY
. show that

ρ
UV

=

ρ
XY

.
EXERCISE 4
Let
U
and
V
be independent random variables with means
μ
and variances
σ
2
. Let
Z
=
αU
+
V
√
1

α
2
.
Find
E
(
Z
) and
ρ
UZ
.
EXERCISE 5
Suppose that
X
and
Y
are two independent measurements. Also it is given that
E
(
X
) =
E
(
Y
) =
μ
, but
σ
X
and
σ
Y
are unequal. The two measurements are combined by means of a weighted average to give
Z
=
αX
+ (1

α
)
Y
where
α
is a scalar and 0
≤
α
≤
1.
a. Show that
E
(
Z
) =
μ
.
b. Find
α
in terms of
σ
X
and
σ
Y
to minimize
V ar
(
Z
).
c. Under what circumstances is it better to use the average
X
+
Y
2
than either
X
or
Y
alone to estimate
μ
. Note:
X,Y,
X
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '07
 Wu
 Statistics, Variance, Probability theory, probability density function, Nicolas Christou

Click to edit the document details