Relative Frequency

 Relative Frequency 

In order to motivate the approach adopted for the solution of the above problem, consider the following procedure. Suppose that we repeat the experiment e n times and let A and B be two events associated with e. We let n_A and n_B be the number of times that the event A and the event B occurred among then repetitions, respectively.

Definition. 

f_A = n_A /n is called the relative frequency of the event A .in the n repetitions of 8. The relative frequency f_A has the following important properties, which are easily verified.
(1) 0 ≤ f_A  ≤ 1
(2) f_A = I if and only if A occurs every time among the n repetitions.
(3) f_A = 0 if and only if A never occurs among the n repetitions.
( 4) If A and B are two mutually exclusive events and if f_A⋃B is the relative frequency associated with the event A⋃ B, then f_A⋃B  = f_A+ f_s.
(5) f_A, based on n repetitions of the experiment and considered as a function of n, "converges" in a certain probabilistic sense to P(A) as n→∞

Note: Property (5) above is obviously stated somewhat vaguely at this point. Only later (Section 12.2) will we be able to make this idea more precise. For the moment let us simply state that Property (5) involves the fairly intuitive notion that the relative frequency based on an increasing number of observations tends to "stabilize" near some definite value. This is not the same as the usual concept of convergence encountered
elsewhere in mathematics. In fact, as stated here, this is not a mathematical conclusion at all but simply an empirical fact. Most of us are intuitively aware of this phenomenon of stabilization although we may never have checked on it. To do so requires a considerable amount of time and patience, since it involves a large number of repetitions of an experiment. However, sometimes we may be innocent observers of this phenomenon as the following example illustrates.

EXAMPLE 1

Suppose that we are standing on a sidewalk and fix our attention on two adjacent slabs of concrete. Assume that it begins to rain in such a manner that we are actually able to distinguish individual raindrops and keep track of whether these drops land on one slab or the other. We continue to observe individual drops and note their point of impact. Denoting the ith drop by  X_i, where X_i= 1 if the drop lands on one slab and 0 if it lands on the other slab, we might observe a sequence such as 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1. Now it is clear that we are not able to predict where a particular drop will fall. (Our experiment consists of some sort of meteorological situation causing the release of raindrops.) If we compute the relative frequency of the event A = {the drop lands on slab I}, then the above sequence of outcomes gives rise to the following relative frequencies (based on the observance of I, 2, 3, . .. drops): 1 , 1 , 1/2 , 2/3 , 3/4 , 3/5 , 3/6 , 3/7 , 4/8 , 4/9 , 4/10 , 4/10 , 5/11 ........................ These numbers show a considerable degree of variation, particularly at the beginning. It is intuitively clear that if the above experiment were continued indefinitely, these relative frequencies would stabilize near the value t. For we have every reason to believe that after some time had elapsed the two slabs would be equally wet. This stability property of relative frequency is a fairly intuitive notion as yet, and we will be able to make it mathematically precise only later. The essence of this property is that if an experiment is performed a large number of times, the relative frequency of occurrence of some event A tends to vary less and less as the number of repetitions is increased. This characteristic is also referred to as statistical regularity. We have also been somewhat vague in our definition of experiment. Just when is a procedure or mechanism an experiment in our sense, capable of being studied mathematically by means of a nondeterministic model? We have stated previously that an experiment must be capable of being performed repeatedly under essentially unchanged conditions. We can now add another requirement. When the experiment is performed repeatedly it must exhibit the statistical regularity referred to above. Later we shall discuss a theorem (called the Law of Large Numbers) which shows that statistical regularity is in fact a consequence of the first requirement: repeatability.