Information Technology Reference
In-Depth Information
Lemma 1
The probability that any one of the elementary events contained within the event space-time will
occur between two successive time points t
0
and t
1
given that the contents/contours of the event space
remains unchanged from t
0
to t
1
is unity i.e. P (U
0
| U
0
= U
1
) = 1. By extension, P(U
t
| U
t
= U
t+1
) =
1 for all t = 0, 1, 2, 3, ...
Proof
Lemma 1 results from a natural extension of Kolmogorov's second axiom if we allow the
event space to be of a time-dynamic nature i.e. if U is allowed to evolve through time in
discrete intervals.
QED
Lemma 2
If the classical probability of occurrence of a specific elementary event E contained within the event
space-time is defined as P(E), then the first-order probability of occurrence of such event E becomes
P{P(E)} = P
1
(E) = P{E
| (U
0
= U
1
)}
=
P(E)
.
[P{(U
0
= U
1
)|E}/P(U
0
= U
1
)]
Proof
Applying the fundamental law of conditional probability we can write as follows:
P{E
| (U
0
= U
1
)} = P{E∩(U
0
= U
1
)}/ P(U
0
= U
1
)
P{E∩(U
0
= U
1
)} = P{(U
0
= U
1
)∩E} = P(E)
.
P{(U
0
= U
1
)|E}; and thus the result follows.
QED
Lemma 3
Given the first-order probability of occurrence of elementary event E and assuming that (U
t
= U
t+1
)
and (U
t+1
= U
t+2
) are independent for all t = 0, 1, 2, 3, ..., the second-order probability of occurrence
of E becomes P
2
(E) = P(E)
.
P
1
(E)
.
[P{(U
1
= U
2
)|E}/P(U
1
= U
2
)].
Proof
By definition, P
2
(E) = P{P
1
(E)} = P[{E | (U
0
= U
1
)} ∩ {E|(U
1
= U
2
)}]
Since (U
t
= U
t+1
) and (U
t+1
= U
t+2
) are assumed independent for t = 0, 1, 2, 3, ..., we can write:
P[{E | (U
0
= U
1
)} ∩ {E|(U
1
= U
2
)}] = P{E | (U
0
= U
1
)}
.
P{E|(U
1
= U
2
)}.
Substituting P{E | (U
0
= U
1
)} with P
1
(E) and then applying the fundamental law of
conditional probability; the result follows.
QED
Thus, given the first-order probability of occurrence of an elementary event E, the second-
order probability is obtained as a “probability of the first-order probability” and is
necessarily
either equal to or less than
the first-order probability, as is suggested by common
intuition. This logic could then be extended to each of the subsequent higher order
probability terms. Based on lemmas 1 - 3, we next propose and prove a fundamental
theorem of higher order (hereafter
H
-
O
) probabilities.
