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CE 3102
Fall 2008
Thought Problem 17
Signal Timing, and Estimating Means
In timing a traffic signal, the duration of a yellow interval should be such that it
eliminates the dilemma zone, where a driver can neither stop without entering the
intersection, nor
clear the intersection before the signal turns red.
The yellow interval (
τ
) which accomplishes this sets the stopping distance
d
s
equal to the
maximum clearing distance d
c
, or
L
W
v
a
v
vt
p
−
−
=
+
2
2
where
v
is vehicle speed,
a
is the braking deceleration,
t
p
is the driver's reaction time,
W
is the width of the cross street and
L
is the vehicle's length.
v
L
W
a
v
t
p
+
+
+
=
2
For a particular intersection, the width of the crossstreet (
W
) is easy to obtain, as are
recommended default values for
reaction time, deceleration, and vehicle length , and it is
often
recommended that the mean approach speed be used for
v
.
A traffic engineer has obtained the following individual vehicle speed measurements (in
mph) on an approach for an intersection:
33, 32, 33, 30, 27, 35, 34, 38, 46, 39
(1)
How should the engineer estimate the mean approach speed?
(2)
Can the engineer be "reasonably certain" that this estimate is within 1 mph of the true
mean speed?
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View Full DocumentCE 3102
Fall 2008
Thought Problem 17
Instructor’s Partial Solution, part 1
(1)
We claim that the sample average
mph
x
n
x
n
i
i
7
.
34
1
1
=
⎟
⎠
⎞
⎜
⎝
⎛
=
∑
=
is a "good" estimate of the underlying mean speed. What does this mean?
First, let's distinguish between an "estimate" which is a real number obtained by
performing the estimating operation on the actual data in our sample, and an "estimator",
which is a random variable obtained by performing the estimating operation on a
sequence of random variables. That is
Observations:
x
1
x
2
….
x
n
Random
Variables:
X
1
X
2
…
X
n
Our sample average as an
estimator
would be
∑
=
⎟
⎠
⎞
⎜
⎝
⎛
=
n
i
i
X
n
X
1
1
We then say an estimate is good if it comes from using a "good" estimator. So what
makes
X
a good estimator?
Suppose that the X
i
are independent, and identically (iid) distributed random variables,
with common mean (expected value)
μ
and common variance
σ
2
. Since
X
is also a
random variable,
it too has an expected value and a variance. More particularly,
∑
∑
=
=
=
=
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
n
i
i
n
i
i
n
n
X
E
n
X
n
E
X
E
1
1
)
(
1
1
)
(
μ
In words, the expected value of the sample average (as a random variable) equals the
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 Spring '11

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