This preview shows pages 1–3. Sign up to view the full content.
EXERCISES IN STATISTICS
Series A, No. 7
1. Let
x
and
y
be jointly distributed random variables with conditional ex
pectations which can be written as
E
(
y

x
)=
α
+
βx
and
E
(
x

y
γ
+
δy
.
Express
β
and
δ
in terms of the moments of the joint distributions and
show that
β
≤
1
/δ
.
Answer.
We have
E
(
y

x
α
+
with
β
=
C
(
x, y
)
V
(
x
)
and
α
=
E
(
y
)

βE
(
x
)
.
Likewise
E
(
x

y
γ
+
with
δ
=
C
(
x, y
)
V
(
y
)
and
γ
=
E
(
x
)

(
y
)
.
On forming the product of the two slope coeﬃcients we can invoke the Cauchy–
Schwarz Inequality:
βδ
=
C
(
x, y
)
2
V
(
x
)
V
(
y
)
=
'
Corr
(
x, y
)
“
2
≤
1
.
Hence
β
≤
1
/δ
.
2. A marksman’s scores are a sequence of random variables
y
i
with
E
(
y
i
)=90
and
V
(
y
i
) = 16 for all
i
. The correlation between successive scores is 0.9,
and the expectation of a score conditional upon the previous score is given
by
E
(
y
i

y
i

1
α
+
βy
i

1
where
α
=(1

β
)
E
(
y
i
). Find the expected
score given that the previous score was 80.
Answer.
Substituting for
α

β
)
E
(
y
i
). gives
E
(
y
i

y
i

1
α
+
i

1

β
)
E
(
y
i
)+
i

1
=
E
(
y
i
β
'
y
i

1

E
(
y
i
)
“
Also, since
V
(
y
i
V
(
y
i

1
), we have
(
y,y
i

1
C
(
y
i
,y
i

1
)
p
V
(
y
i
)
V
(
y
i

1
)
=
C
(
y
i
i

1
)
V
(
y
i

1
)
=
β.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document SERIES A No.7, ANSWERS
With
E
(
y
i
)=
E
(
y
i

1
) = 90,
Corr
(
y
i
,y
i

1
)=9
/
10, and
y
i

1
= 80, this gives
E
(
y
i

y
i

1
)=90+
9
10
(80

90)=81
.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 03/02/2012 for the course EC 2019 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.
 Spring '12
 D.S.G.Pollock

Click to edit the document details