Problem 1
(a)
L
is a uniform random variable on the interval [0
,
10], so
E
[
L
] =
Z
10
0
t
(1
/
10)
dt
= 5
.
(b) Clearly,
E
[
L
2
] =
Z
10
0
t
2
(1
/
10)
dt
= 100
/
3
so we see that the variance of
L
is just
V ar
(
L
) = 100
/
3

100
/
4 = 100
/
12 = 25
/
3
.
(c) The c.d.f. of
L
is the function
F
(
t
), for
t
∈ <
(this must be computed for
all
t
). Notice that when
t <
0,
F
(
t
) =
Z
t
∞
f
(
s
)
ds
= 0
.
When 0
≤
t <
1,
F
(
t
) =
Z
t
∞
f
(
s
)
ds
=
Z
t
0
(1
/
10)
ds
=
t/
10
.
, Finally, when
t
≥
1,
F
(
t
) =
Z
t
∞
f
(
s
)
ds
=
Z
1
0
(1
/
10)
ds
= 1
.
Putting everything together, we see that the cdf of
L
is just
F
(
t
) =
0
,
t <
0;
t/
10
,
0
≤
t <
10;
1
,
t
≥
10.
(d) The round trip travel time
R
consists of walking to the item, picking up the
item, and walking back to the initial point. Therefore,
R
=
L/
2+3+
L/
2 =
L
+3
(e) The mean of
R
is just
E
[
R
] =
E
[
L
+ 3] = 8.
(f) The variance of
R
is just
V ar
(
R
) =
V ar
(
L
) = 25
/
3.
(g) The c.d.f. of
R
(which we will denote as
F
R
) is just
F
R
(
t
) =
0
,
t <
3;
(
t

3)
/
10
,
3
≤
t <
13;
1
,
t
≥
13.
1
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 Spring '08
 Zahrn
 Variance, Probability theory, Exponential distribution, L∗

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