Yurii Kosovtsov
Oct3-04, 03:49 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\nThe probability theory is very nice logically consistent theory. The\nhallmark of the theory is the notion of (statistical) independence.\nIt is very essential that for each problem in the framework of the\nprobability theory the initial (probability) measure is specified.\n\nBut in the physical statement of probabilistic problems we face some\nfundamental difficulties.\n\n1. One of the basic concepts of physics is the use only measurable\nvariables. In the sense, that there are instrumentation and procedure\nto obtain a "number" for given variable. Alas, there is not any\nprocedure to obtain the "probabilistic number" without hiring a priori\nthe (statistical) independence for series of mensurations. That is we\nobviously fall into a circle. It is not surprising that very often the\nprobabilistic characteristics of input processes of some physical\nsystems are considered as uncertain.\n\n2. This forces to formulate "physical" ("intuitive") versions of\n(statistical) independence, which dramatically differs from its\noriginal mathematical meaning. In most popular version two random\nvariables are considered as (statistically) independent if they are\naffected by many causally independent disturbances. The more or less\nrigorous consideration of physical systems from the probability theory\npoint of view makes it clear that the probability to meet\n(statistically) independent physical random variables is equal to\nzero.\n\n3. Spurious way out of points 1 and 2 commonly is considered as\nfollows. Yes, the probabilistic characteristics of input processes of\nsome physical systems are uncertain. But the probability theory has\nthe extremely helpful proposition - the Central Limit Theorem (CLT).\n\nThe output processes are usually a result of an inertial\ntransformation of initial random functions, therefore very often they\nmay be regarded as a sum of a large number of random items. For this\nreason and owing to CLT we as though come to a conclusion that for\nmany practically important cases the distribution function of the\noutput of the linear (or even non-linear) inertial system is\napproaches to Gaussian one. Hence it follows very powerful\nconsequences: despite the fact that a probability as a measure on a\ncertain set of elementary events is a priori unknown, the profound\nconclusions are possible, since for many cases the functions\ninterested us have been determined on this set are weakly depend on\nits exact probability measure. In a different way, as a result of a\ntransformation the probability measure is unified and practically\n"forgets" about own origin. This position is extremely attractive\nsince it causes an impression that it is possible to extract "free of\ncharge" the probability laws from anything. It is unlikely possible\nto embrace all fields of the probability theory applications where\nGaussian distribution of a random function is led "naturally" only on\nreason of inertial character of a dependence of a "output" random\nfunction from "input" one or when the outcome of a chance experiment\nis determined by a large number of random factors. All of these had\ncreated the ground for deifying of Gaussian distribution.\n\nHowever, the main feature of totality of statements are known under\nthe name CLT of the probability theory is that under increasing of\nnumber of items the distribution of sums of (statistically)\nindependent items is tending to Gaussian one IF AND ONLY IF the sum is\nNORMALIZED by STRICTLY definite way.\n\nBut in the nature there is not this specific however necessary\nnormalization. There is not proper counter of numbers N of independent\nitems in any physical systems and there is not proper divider on (N)^1/2. That\nis why there are not grounds for CLT application here.\n\nSo, what is the physical meaning of the probability theory?\n\nYurii Kosovtsov\nLviv Radio Engineering Research Institute, Ukraine\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>The probability theory is very nice logically consistent theory. The
hallmark of the theory is the notion of (statistical) independence.
It is very essential that for each problem in the framework of the
probability theory the initial (probability) measure is specified.
But in the physical statement of probabilistic problems we face some
fundamental difficulties.
1. One of the basic concepts of physics is the use only measurable
variables. In the sense, that there are instrumentation and procedure
to obtain a "number" for given variable. Alas, there is not any
procedure to obtain the "probabilistic number" without hiring a priori
the (statistical) independence for series of mensurations. That is we
obviously fall into a circle. It is not surprising that very often the
probabilistic characteristics of input processes of some physical
systems are considered as uncertain.
2. This forces to formulate "physical" ("intuitive") versions of
(statistical) independence, which dramatically differs from its
original mathematical meaning. In most popular version two random
variables are considered as (statistically) independent if they are
affected by many causally independent disturbances. The more or less
rigorous consideration of physical systems from the probability theory
point of view makes it clear that the probability to meet
(statistically) independent physical random variables is equal to
zero.
3. Spurious way out of points 1 and 2 commonly is considered as
follows. Yes, the probabilistic characteristics of input processes of
some physical systems are uncertain. But the probability theory has
the extremely helpful proposition - the Central Limit Theorem (CLT).
The output processes are usually a result of an inertial
transformation of initial random functions, therefore very often they
may be regarded as a sum of a large number of random items. For this
reason and owing to CLT we as though come to a conclusion that for
many practically important cases the distribution function of the
output of the linear (or even non-linear) inertial system is
approaches to Gaussian one. Hence it follows very powerful
consequences: despite the fact that a probability as a measure on a
certain set of elementary events is a priori unknown, the profound
conclusions are possible, since for many cases the functions
interested us have been determined on this set are weakly depend on
its exact probability measure. In a different way, as a result of a
transformation the probability measure is unified and practically
"forgets" about own origin. This position is extremely attractive
since it causes an impression that it is possible to extract "free of
charge" the probability laws from anything. It is unlikely possible
to embrace all fields of the probability theory applications where
Gaussian distribution of a random function is led "naturally" only on
reason of inertial character of a dependence of a "output" random
function from "input" one or when the outcome of a chance experiment
is determined by a large number of random factors. All of these had
created the ground for deifying of Gaussian distribution.
However, the main feature of totality of statements are known under
the name CLT of the probability theory is that under increasing of
number of items the distribution of sums of (statistically)
independent items is tending to Gaussian one IF AND ONLY IF the sum is
NORMALIZED by STRICTLY definite way.
But in the nature there is not this specific however necessary
normalization. There is not proper counter of numbers N of independent
items in any physical systems and there is not proper divider on (N)^1/2. That
is why there are not grounds for CLT application here.
So, what is the physical meaning of the probability theory?
Yurii Kosovtsov
Lviv Radio Engineering Research Institute, Ukraine
hallmark of the theory is the notion of (statistical) independence.
It is very essential that for each problem in the framework of the
probability theory the initial (probability) measure is specified.
But in the physical statement of probabilistic problems we face some
fundamental difficulties.
1. One of the basic concepts of physics is the use only measurable
variables. In the sense, that there are instrumentation and procedure
to obtain a "number" for given variable. Alas, there is not any
procedure to obtain the "probabilistic number" without hiring a priori
the (statistical) independence for series of mensurations. That is we
obviously fall into a circle. It is not surprising that very often the
probabilistic characteristics of input processes of some physical
systems are considered as uncertain.
2. This forces to formulate "physical" ("intuitive") versions of
(statistical) independence, which dramatically differs from its
original mathematical meaning. In most popular version two random
variables are considered as (statistically) independent if they are
affected by many causally independent disturbances. The more or less
rigorous consideration of physical systems from the probability theory
point of view makes it clear that the probability to meet
(statistically) independent physical random variables is equal to
zero.
3. Spurious way out of points 1 and 2 commonly is considered as
follows. Yes, the probabilistic characteristics of input processes of
some physical systems are uncertain. But the probability theory has
the extremely helpful proposition - the Central Limit Theorem (CLT).
The output processes are usually a result of an inertial
transformation of initial random functions, therefore very often they
may be regarded as a sum of a large number of random items. For this
reason and owing to CLT we as though come to a conclusion that for
many practically important cases the distribution function of the
output of the linear (or even non-linear) inertial system is
approaches to Gaussian one. Hence it follows very powerful
consequences: despite the fact that a probability as a measure on a
certain set of elementary events is a priori unknown, the profound
conclusions are possible, since for many cases the functions
interested us have been determined on this set are weakly depend on
its exact probability measure. In a different way, as a result of a
transformation the probability measure is unified and practically
"forgets" about own origin. This position is extremely attractive
since it causes an impression that it is possible to extract "free of
charge" the probability laws from anything. It is unlikely possible
to embrace all fields of the probability theory applications where
Gaussian distribution of a random function is led "naturally" only on
reason of inertial character of a dependence of a "output" random
function from "input" one or when the outcome of a chance experiment
is determined by a large number of random factors. All of these had
created the ground for deifying of Gaussian distribution.
However, the main feature of totality of statements are known under
the name CLT of the probability theory is that under increasing of
number of items the distribution of sums of (statistically)
independent items is tending to Gaussian one IF AND ONLY IF the sum is
NORMALIZED by STRICTLY definite way.
But in the nature there is not this specific however necessary
normalization. There is not proper counter of numbers N of independent
items in any physical systems and there is not proper divider on (N)^1/2. That
is why there are not grounds for CLT application here.
So, what is the physical meaning of the probability theory?
Yurii Kosovtsov
Lviv Radio Engineering Research Institute, Ukraine