IE 230
Seat # ________
Name ___ < KEY > ___
Closed book and notes. 60 minutes.
Cover page and four pages of exam.
Pages 8 and 12 of the Concise Notes.
No calculator. No need to simplify answers.
This test is cumulative, with emphasis on Section 4.8
through Section 5.5 of Montgomery and Runger, fourth edition.
Remember: A statement is true only if it is always true.
One point: On the cover page, circle your family name.
One point: On every page, write your name.
The random vector (
X
1
,
X
2
, . . . , X
k
) has a
multinomial distribution
with joint pmf
P(
X
1
=
x
1
,
X
2
=
x
2
, . . . , X
k
=
x
k
)
=
x
1
!
x
2
!
. . .
x
k
!
n
!
hhhhhhhhhhhhh
p
1
x
1
p
2
x
2
. . .
p
k
x
k
when each
x
i
is a nonnegative integer and
x
1
+
x
2
+
. . .
+
x
k
=
n
; zero elsewhere.
The linear combination
Y
=
c
0
+
c
1
X
1
+
c
2
X
2
+
. . .
+
c
n
X
n
has mean and variance
E(
Y
)
=
c
0
+
i
=
1
Σ
n
E(
c
i
X
i
)
=
c
0
+
i
=
1
Σ
n
c
i
E(
X
i
)
and
V(
Y
)
=
i
=
1
Σ
n
j
=
1
Σ
n
cov(
c
i
X
i
,
c
j
X
j
)
=
i
=
1
Σ
n
j
=
1
Σ
n
c
i
c
j
cov(
X
i
,
X
j
)
=
i
=
1
Σ
n
c
i
2
V
(
X
i
)
+
2
i
=
1
Σ
n
−
1
j
=
i
+
1
Σ
n
c
i
c
j
cov(
X
i
,
X
j
)
.
Cov(
X
,
Y
)
=
E[(
X
− μ
X
) (
Y
− μ
Y
)]
Corr(
X
,
Y
)
=
Cov(
X
,
Y
)
/
(
σ
X
σ
Y
)
If (
X
,
Y
) is bivariate normal, then
X
and
Y
are normal. In addition, the conditional
distribution of
X
given that
Y
=
y
is normal, with mean
μ
X
+ ρ
X
,
Y
σ
X
[(
y
−μ
Y
)
/
σ
Y
] and
variance (1
− ρ
X
,
Y
2
)
σ
X
2
.
Score ___________________________
Exam #3, April 12, 2011
Schmeiser