Given to the “probability” of life on Mars, for which there can be
No frequency.) What was said in S.7/4 is relevant here, for the
Concept of probability is, in its practical aspects, meaningful only
Over some set in which the various events or possibilities occur
With their characteristic frequencies.
The test for a constant probability thus becomes a test for a con-
Stant frequency. The tester allows the process to continue for a
Time until some frequency for the event has declared itself. Thus,
If he wished to see whether Manchester had a constant, i.e.
Unvarying, probability of rain (in suitably defined conditions), he
Would record the rains until he had formed a first estimate of the
Frequency. He would then start again, collect new records, and
163
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
I N CESSA N T TR AN SMI SSIO N
Form a second estimate. He might go on to collect third and fourth
Estimates. If these several estimates proved seriously discrepant
He would say that rain at Manchester had no constant probability.
If however they agreed, he could, if he pleased, say that the frac-
Tion at which they agreed was the constant probability. Thus an
Event, in a very long sequence, has a “constant” probability of
Occurring at each step if every long portion of the sequence shows
It occurring with about the same relative frequency.
These words can be stated more accurately in mathematical
Terms. What is important here is that throughout this book any
Phrases about “probability” have objective meanings whose
Validity can be checked by experiment. They do not depend on
Any subjective estimate.
Ex. 1: Take the five playing cards Ace, 2, 3, 4, 5. Shuffle them, and lay them in
A row to replace the asterisks in the transformation T:
Ace 2345T: ↓ *****
Is the particular transformation so obtained determinate or not? (Hint: Is it
Single-valued or not?)
Ex. 2: What rule must hold over the numbers that appear in each column of a
Matrix of transition probabilities?
Ex. 3: Does any rule like that of Ex. 2 hold over the numbers in each row?
Ex. 4: If the transformation defined in this section starts at 4 and goes on for 10
|
|
|
Steps, how many trajectories occur in the set so defined?
Ex. 5: How does the kinematic graph of the stochastic transformation differ from
That of the determinate ?
Of the type we have considered throughout the book till now. The
Single-valued, determinate, transformation is thus simply a spe-
Cial, extreme, case of the stochastic. It is the stochastic in which
All the probabilities have become 0 or 1. This essential unity
Should not be obscured by the fact that it is convenient to talk
Sometimes of the determinate type and sometimes of the types in
Which the important aspect is the fractionality of the probabilities.
Throughout Part III the essential unity of the two types will play
An important part in giving unity to the various types of regulation.
The word “stochastic” can be used in two senses. It can be used
To mean “all types (with constant matrix of transition probabili-
Ties), the determinate included as a special case”, or it can mean
“all types other than the determinate”. Both meanings can be
Used; but as they are incompatible, care must be taken that the
Context shows which is implied.
T HE M AR KOV CHAI N
After eight chapters, we now know something about how a sys-
Tem changes if its transitions correspond to those of a singlevalued
Transformation. What about the behaviour of a system whose tran-
Sitions correspond to those of a stochastic transformation? What
Would such a system look like if we met one actually working?
Suppose an insect lives in and about a shallow pool— some-
Дата добавления: 2019-11-16; просмотров: 223; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!