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
!
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
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IE 230 — Probability & Statistics in Engineering I
Name ___ < KEY > ___
Closed book and notes. 60 minutes.
1. (3 points each) True or false.
Consider the notes from the cover page.
(a)
T
F
←
E(
X
−
Y
)
=μ
X
+μ
Y
.
(b)
T
←
F
Corr(
X
,
Y
)
≥
0 implies that Cov(
X
,
Y
)
≥
0.
(c)
T
F
←
If (
X
,
Y
) is bivariate normal, then
ρ
X
,
Y
=
0.
(d)
T
F
←
If
X
1
and
X
2
are independent and exponentially distributed, then
X
1
+
X
2
has an exponential distribution.
(e)
T
←
F
If (
X
,
Y
) is bivariate
normal,
then
4.5(
X
+
Y
) is normally
distributed.
(f)
T
F
←
If
v
and
w
are real numbers, then
f
X
,
Y
(
v
,
w
)
=
f
X
(
v
)
f
Y
(
w
).
(g)
T
←
F
If
X
and
Y
are independent, then
ρ
X
,
Y
=
0.
(h)
T
←
F
The time to recharge a battery is never normally distributed.
2. (3 points each) Consider the notation from the cover page. For each of the following,
indicate whether the expression is a constant, an event, a random variable, or undefined.
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