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Szekely G.J. Paradoxes in probability theory and mathematical statistics (D.Reidel, 1986)(KA)(T)(264s)_MV_.djvu |
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Xx, X2, ..., Xn are uniformly distributed on the interval C; 23), the maxi-
maximum likelihood estimator of 3 is
TJ — — ma-sr (Y Y Y 1
2
and a slight modification leads to the unbiased estimator
The following estimator A is also unbiased but with smaller variance...
The essence
of the paradox is the following: it is possible that observations which
have nothing to do with a parameter can influence its good estimations
(cf...
Thus many "stable" ("robust") and
multivariate methods with an enormous quantity of operations entered
the practice of everyday statistics...
Let the generating function of the
probability distribution po,Pi,p2, ••• be denned by
( Z
where |z|sl...
(See Mandelbroit's
book.)
(if) In his book Mandelbroit also mentions other notions of dimen-
dimension, such as the Fourier dimension...
The time between
the arrival of the first and («+l)th car is
B-l
where
If
M@=max{«: J Yt ^ t},
then the number of cars which can get through the green light from time
t to t+h is M{t+h)—M{t)...
Let pk denote the probability of moving to the right and
1 — pk the probability of moving to the left...
b) The paradox
If a share is expected to be profitable, it seems natural that the share is
worth buying, and if it is not profitable, it is worth selling...
Let us denote the average number of Jacob's and Laban's sheep in
the nth year, respectively, by /„ and Ln (in the initial, Oth year /0=0
and Lo is a positive number)...
It is also surprising that the probability that one
team leads throughout the second half is 50 per cent, no matter how
large n is...
As the ratio Pm.a:(lvi+a is independent of &, the condition-
166 Chapter 3
al distributions P2k+a/2 Pu+a an£i 92k+a/2 tfat+a are identical—as we
have stated...
In 1950 Kenneth Arrow (the 1972 Nobel Prize winner in
economics) used the above example to show that it is logically impossible
to create an absolutely fair election system...
The question arises: When defining probability, why do we need sigma-
algebras instead of the set of all the subsets of phase space £2? The answer
is very simple: In general, the probability cannot be defined on the set
of all the subsets of Q, more precisely, if probability is defined on a sigma-
algebra consisting of some subsets of Q, then this probability may not be
extended to the rest of the subsets of Q if sigma-additivity is still required
(unless Q consists of finite or countably infinite elements)...
If the limit of the relative frequency kjn when n
tends to infinity exists then this limit is called the probability of K...
However, if we assume
only additivity, (that is, the measure of the union of two disjoint sets
equals the sum of their individual measures), then—as the Polish mathe-
Paradoxes in the foundations of probability theory 181
matician S...
The Banach—Tarski paradox shows that changing
sigma-additivity for additivity does not solve every problem and also
brings new ones...
It is obvious therefore that the Monte Carlo method only
became applicable when experiments could be simulated by computers...
b) The paradox
In most numbers digits follow each other randomly, that is, most of
the numbers are uninteresting in the following sense: the computer
programs which produce these numbers are not much shorter than the
numbers themselves...
In connec-
connection with these infinite graphs, Erdos and Renyi drew attention to the
following paradox...
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