Digital Signal Processing Reference
In-Depth Information
Is it correct to say that the events
A
1
,
A
2
,
A
3
are mutually independent?
Q.1.13. Can the union of two events
A
and
B
be equal to their intersection?
Q.1.14. What is the usefulness of Bayes' rule?
1.10 Answers
A.1.1. The reasons may be the following [DAV87, p. 8]:
We do not have enough data.
We do not know all the causal forces at work.
The forces are so complicated that the computation of an exact outcome
will be so complicated and, thus, not useful.
There is some basic indeterminacy in the physical word.
A.1.2. A sample space is not unique and the choice of a particular sample space
depends on the desired result we would like to obtain by performing an
experiment [HAD06, p. 21], [MIL04, p. 11]. For example, in rolling a die
we may be interested only in the even number of dots. In that case, the
outcomes are: 2, 4, and 6, and the sample space is
S ¼
{2, 4, 6}.
A.1.3. Elements in a sample space must always be mutually exclusive because the
particular outcome in an experiment excludes the occurrence of another.
A.1.4. In practical experiments, the characteristics of outcomes may be of inter-
est. Events are sets of outcomes that meet common characteristics.
A.1.5. Not every event with a probability of zero is impossible. For example, in
the die rolling experiment we are interested only in an even number of dots.
Thus, the sample space is
S ¼
{
s
1
,
s
2
,
s
3
}, where
s
1
¼
2,
s
2
¼
4, and
s
3
¼
6.
Let us now define the following events in the sample space
S
:
A ¼f
1
< s
i
<
6
g;
B ¼fs
i
<
2
g;
C ¼fs
i
>
6
g:
(1.131)
Event
A
is:
A ¼
{
s
1
,
s
2
,}.
Event
B
is a null event,
B ¼
{} (i.e., it does not contain any element in
S
).
Therefore, its probability is zero,
P
{
B
}
¼
0. However, it is not an impossi-
ble event because in the die rolling experiment an outcome 1 (i.e.,
s
i
<
2) is
not an impossible outcome.
Conversely, event
C
is an impossible null event (an outcome greater
than 6 cannot occur).
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