ISyE 2027
Probability with Applications
Fall 2011
R. D. Foley
Homework 6
October 17, 2011
due Monday
1. Suppose
X
has p.d.f.
f
(
t
) =
2
t
for
0
6
t
6
1
. Compute (a)
Pr
{
1
/
4
6
X <
3
/
4
}
,
(b)
E
[
X
]
, (c)
E
[
X
2
]
, (d) the variance of
X
, and (e) the c.d.f. of
X
.
2. Suppose
Z
has p.d.f.
f
(
s
) =
ce

5
s
. Compute (a) c, (b)
Pr
{
Z
6
2.5
}
,
(c)
Pr
{
Z >
2.5
}
, (d) the c.d.f. of
Z
, (e)
Pr
{
Z > t
}
for all
t
>
0
, and (f)
Pr
{
Z > t
+
s

Z > t
}
for all
t
,
s
>
0
.
3. Suppose
Y
is a random variable with mean
μ
and variance
σ
2
<
∞
.
What is the mean and variance of
(
Y

μ
)
/σ
.
4. Suppose a person has to pick an item from an aisle that is 50 feet in
length. The person is standing at one end of the aisle. The item is a
random distance
L
feet down the aisle from the picker. The r.v.
L
has
p.d.f.
f
(
t
) =
c
for
0
6
t
6
50
. Determine (a)
c
, (b) the c.d.f. of
L
, (c)
the mean of
L
, and (d) the variance of
L
. Assume that the person walks
at 4 feet per second and takes an additional 10 seconds to pull the item
out of its location. Let
R
be the total time for the person to walk down
to location
L
, pull the item, and then walk back to the front of the aisle.
(e) Express
R
as a function of
L
. (f) Compute the mean of
R
, and (g) the
variance of
R
.
5. An earlier homework assignment contained the following:
Please read the instructions carefully, before doing this problem. After
reading and understanding this problem, ﬂip two coins: a nickel and a
quarter. If the nickel came up heads, answer Question A. If the nickel
came up tails, answer question B.
Question A: Did the quarter come up heads?
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 Zahrn
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