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Selected papers of Richard von Mises (Vol. 2, Probability and Statistics, General) (1964)(600dpi)(T)(577s).djvu |
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Schranken oder eine von ihnen ins Unendliche riicken...
Liegen die Wachstumstellen im Intervall C l bis c 2 , so gilt fur nicht
abnehmendes I" (x)
s2/(c l )+(c l - a)21 ( a + 52 )
a l - c l
S2 + (a - C 1 )2
(16")
5 2 f(c 2 ) + (c 2 - a)2f (a _ 52 )
< @,(f) < C 2 - a
- - S2 + (c 2 - a)2
und fiir nicht zunehmendes I" (x) das Umgekehrte...
Hat man aber
a < 0, so liegt der Beriihrungspunkt x = a wegen I' (a) < 0 auf dem
absteigenden Ast der Parabel, der Schnittpunkt x = c wegen I(c) > I(a)
auf dem aufsteigenden...
Wir wollen hier noch, unter Beniitzung der allgemeinen Formeln, den
Fall m = 3 kurz behandeln...
n s!
Fur das folgende ist wichtig, zu wissen, dass (5) auch unter allge-
meineren Bedingungen gilt...
Endlich hat
man in (9) statt ( ; ) auf der linken Seite und statt den( ; ) auf der rechten
(:) bezw...
Die Gesamtheit der ganzzahligen,
nicht-negativen Losungen von (23) bildet, geometrisch gesprochen, ein
(k - 2)-dimensionales Punktgitter im k-dimensionalen Raum...
Wir wissen aber, das b o gegen
UBER AUFTEILUNGS- UND BESETZUNGSWAHRSCHEINLICHKEITEN 333
ne-a...
On the other hand we claim that our theory, which serves to describe
observable facts, satisfies all reasonable requirements of logical consistency
and is free from contradictions and obscurities...
Another objection, often repeated, reads that there is a contradic-
tion between the existence of a frequency limit and the so-called Bernoulli
theorem which states that sequences of any length showing a frequency of
1, say, can also occur in a case for which the probability equals 1...
For the described
derivation of X' from X we have shown that to any place selection S
which affects the distribution in X' corresponds a certain 51 which, when
applied to X, changes the distribution of X...
In order to define the
probability in this case we have to choose a subset A of the label set and
to count among the first n elements the number n A of those elements the
attributes of which fall into A...
In this way
it turns out to be almost certain that the limiting frequency of the
outcome six in the case of a correct die is i...
which supplies for every given X a smallest value 0 1 and a greatest
va]lle o.2' The definition of our third kollektiv leads to the conclusion: If
(6)
(4) J...
ON THE FOUNDATIONS OF PROBABILITY AND STATISTICS
353
f X2 ('I9 0 )
P(x I (}) dx == F({}, (}o) = min...
It has been
shown that the asymptotic expression for PI ({) I x) whatever Po({}) may
be, is
(14)
PI (D Ix) '" 1 / n e-ln( -"')'/"'(I-"')...
In one mathematical treatment frequency notions first appear
in the theorems, whereas in the other they first appear in the axioms;
but they appear in both...
In closing, it seems proper to call attention to what the measure ad-
vocates consider the real services and contributions of the approach of
Professor von Mises...
In order to avoid the above-mentioned equivocation the author had
suggested a long time ago (1) to call the probabilities corresponding to
P(O) and P a;(O) respectively the initial and the final probability...
A much better founded
objection against the practical application of Bayes' formula consists in
saying that in most cases we have no sufficient information about the func-
tion P(O)...
Using the numbers N l , Nfl;' N1fl; introduced in section 2 the probability Q(x)
is defined by equation (6)...
But in most practical cases
nothing or very little is known about the parameter distribution...
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