Lecture 6
Events and the
Field of Events
The structure of this course follows a natural progression. We begin with the concept of a random variable,
since it is an action that results in numbers. If the action is sufficiently known, then one can identify the
collection of numbers that may possibly result prior to even taking the action. If the action is not sufficiently
known, then it may or may not be able to infer properties of this collection from data that has been obtained by
performing the action.
Example
1
Suppose that the action that will be taken is to measure a person’s EEG for a period of 500 seconds
at a sampling rate of 20 samples per second. Hence, this action corresponds to a random variable whose
dimension is (20 samples/second)(500 seconds) = 10,000 samples (or numbers). Suppose further, that we know
that any number cannot exceed
5
±
volts, since the amplifier is limited to that range. Finally, suppose that we
have 32bit resolution. At this fine level of resolution we can assume that essential any number in the interval
]
5
,
5
[

volts is possible. And so, without even performing the measurement of EEG, we know that the sample
space for random variable
X
=
)
,
,
,
(
000
,
10
2
1
X
X
X
is
S
X
=
}
]
5
,
5
[

)
,
,
,
{(
000
,
10
2
1
k
every
for
x
x
x
x
k

∈
.
Now, suppose that the range of values is not known, and for a given person we have the EEG measurement
shown in the top plot of Figure 1below.
0
50
100
150
200
250
300
350
400
450
500
0.5
0
0.5
EEG
EEG Ch.47
Figure 1.
The plot shows a 500second measurement of EEG recorder from electrode #47. The sample rate is
20 samples/second.
From this plot, one might be tempted to assume that the sample space for any element of
X
is
]
5
.
0
,
5
.
0
[

volts.
Note, however, that this is only one sample of
X
. Consequently, it might not capture the entire range of values
that any
X
k
can take on. At this point one should either consult with the person who took the measurements to
verify that, indeed, the permissible range
is
]
5
.
0
,
5
.
0
[

volts, or presume that the range is larger. If one has no
idea how large it might be, then the safest assumption would be
)
,
([
∞
∞
. □
Example
2
Suppose that you have assigned a problem to 10 students in the class, and that the answer is either
correct or wrong (i.e. there is no partial credit. The act of recording the scores of the 10 students is
X
=
)
,
,
,
(
10
2
1
X
X
X
is
S
X
=
}
}
1
,
0
{

)
,
,
,
{(
10
2
1
k
every
for
x
x
x
x
k
∈
. Now, suppose that you are to report the
average score to the principal. The act of computing this score is
X
X
k
k
∆
=
=
∑
10
1
10
1
. [Note: the symbol
∆
=
ia a
defined
equality; that is, the symbol
X
is defined by the operation associated with it in this case, the averaging
operation.]
Q: What is
X
S
?
A:
X
S
=
______________________________________________________