NTHU MATH 2810
Final Examination Solution
Jan 8, 2008
1. (
16pts, 2pts for each
)
(a) True.
(b) False. The values of a pdf can be larger than 1. However, the integration of a pdf
over any region must have values between 0 and 1.
(c) False. It must be a onetoone transformation.
(d) True.
(e) True.
(f) True.
(g) False. If
X
and
Y
are
independent
, then
E
(
X

Y
) =
E
(
X
). Zero correlation (i.e.,
uncorrelated) is a weaker condition than independence. It cannot guarantee this
property.
(h) False. When
X
and
Y
are independent,
E
(
X/Y
) =
E
(
X
)
E
(1
/Y
)
6
=
E
(
X
)
/E
(
Y
)
in general.
2. (
15pts, 3pts for each
)
(a) Normal(
μ,σ
2
) with
μ
= 68 and
σ
2
= 2.
(b) Gamma(
α,λ
) with
α
= 1000 and
λ
= 5. (An alternative answer that is acceptable
is Exponential(
λ
) with
λ
=
1
1000
/
5
=
1
200
.
)
(c) Poisson(
λ
) with
λ
= 2
×
2 = 4.
(d) Binomial(
n,p
) with
n
= 20 and
p
= 1
/
8.
(e) Uniform(
a,b
) with
a
= 0 and
b
= 360.
3. (
6pts
) Let
X
be the number of “5” that occurs in the 500 rolls, then
X
∼
Binomial(500
,
1
/
6)
since the die is fair. Therefore,
E
(
X
) = 500
×
(1
/
6) = 500
/
6
,
and
V ar
(
X
) = 500
×
(5
/
36) = 2500
/
36
.
Because
n
= 500 is large, we can use Normal approximation to evaluate
P
(
X
≥
100) as
follows:
P
(
X
≥
100) =
P
(
X
≥
99
.
5) =
P
X

500
6
q
2500
36
≥
99
.
5

500
6
q
2500
36
≈
P
Z
≥
99
.
5

500
6
q
2500
36
= 1

Φ
99
.
5

500
6
q
2500
36
= 1

Φ(1
.
94)
,
where
Z
∼
Normal(0, 1).
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document4. (a) (
2pts
) Because
X
∼
Uniform(0
,L/
2),
Y
∼
Uniform(
L/
2
,L
), and they are indepen
dent, their joint pdf is
f
X,Y
(
x,y
) =
f
X
(
x
)
f
Y
(
y
) =
(
1
L/
2
1
L/
2
=
4
L
2
,
for 0
< x < L/
2 and
L/
2
< y < L,
0
,
otherwise.
(b) (
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 ShaoWeiCheng
 Math, Probability, Trigraph, joint PDF, ij

Click to edit the document details