STA 4321
Sample Problems for Exam 3
Fall 1999
Mathematical Statistics I
These questions are only meant as a study aid and to help you test your knowledge. Being able to solve them does not
guarantee that you are wellprepared for the exam.
1.
For each of the following joint densities, indicate whether
Y
1
and
Y
2
are independent (Yes or No). No explanation is required.
(a)
f
(
y
1
,y
2
) =
±
2
,
0
≤
y
1
≤
1
,
0
≤
y
2
≤
1
,
0
≤
y
1
+
y
2
≤
1
,
0
,
elsewhere.
(b)
f
(
y
1
,y
2
) =
±
e

(
y
1
+
y
2
)
, y
1
≥
0
,y
2
≥
0
,
0
,
elsewhere.
(c)
f
(
y
1
,y
2
) =
±
y
1
+
y
2
,
0
≤
y
1
≤
1
,
0
≤
y
2
≤
1
,
0
,
elsewhere.
(d)
f
(
y
1
,y
2
) =
±
e

y
1
, y
1
≥
0
,
0
≤
y
2
≤
1
,
0
,
elsewhere.
(e)
f
(
y
1
,y
2
) =
±
e

y
1
,
0
≤
y
2
≤
y
1
<
∞
,
0
,
elsewhere.
2.
Let
Y
1
and
Y
2
denote the proportion of time, out of one workday, that employees I and II, respectively, actually spend
performing their assigned tasks. The joint relative frequency behavior of
Y
1
and
Y
2
is modeled by the density function
f
(
y
1
,y
2
) =
±
y
1
+
y
2
,
0
≤
y
1
≤
1;0
≤
y
2
≤
1
0
,
elsewhere.
(a) Find
P
(
Y
1
≥
1
2
,Y
2
≥
1
2
)
.
(b) Find the marginal density function for
Y
2
.
(c) Find
P
(
Y
2
≥
1
2
)
.
(d) Find