Материал: Крючков Фундаменталс оф Нуцлеар Материалс Пхысицал Протецтион 2011
universe was. Problems of the kind form the subject matter of MS. It may be said: use of PT, with the nature of a phenomenon being known, helps us find out how some characteristics of interest observable in an experiment will behave (are distributed). It is vice versa in statistics where the input is experimental data (these are normally observations of random quantities), and the requirement is to derive a statement (or decide) on the identity of the phenomenon under study.
Probability
Probability of event is generically comprehended as an abstraction that originates in the idea of the relative frequency at which an event occurs in the sequence of a replicated experiment (test) with the “given set of conditions”.
Imagine a relative frequency of an event in an unlimitedly protracted series of a replicated experiment, and you will get close to giving a reasoned interpretation of the probability of an event. Complexities will however arise if you try to base PT on strictly formalizing such interpretation. A simpler and deeper formulation was proposed by A.N. Kolmogorov [2]. It relies on the following assumptions:
∙interval numbers [0,1] may be chosen as the probability of an event in a certain initial class of relatively simple events;
∙these initial probabilities, along with the axioms and rules to determine the probabilities of more complex events, help determine the probability of any event.
Note. Therefore, choosing the probability of an event is normally governed by hypothetical data based on predictions of relative frequencies. One may ask if we have chosen correctly the probability in a given problem. Formally, this question leads us to a problem in theory of testing statistical hypotheses. If solved, this problem brings us to an answer on how to choose probability. Let us introduce these axioms and rules.
1.For every event А there is a given number Р(А) satisfying the condition 0 ≤ P(A) ≤ 1 , which is called probability.
2.The probability of a credible event is equal to unity.
3.The probability of the sum of mutually exclusive events is equal to the sum of the probabilities thereof (the so-called axiom of addition of probabilities).
Let us note again that the system of axioms does not determine the principle of choosing the probability’s numerical value. It is found such
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that all axioms be fulfilled and the laws to which the event occurs be described.
Probability is an objective measure and exists (can be known!) a priori, while relative frequency is always empirical.
We will treat event А as not dependent on event В if the probability of event А does not change when event В occurs. So the pairwise independence we have introduced can be generalized to an independence in a set for a reasonably broad series of events.
Total probability formula and Bayesian formula
The probability of event F that occurs only with each of the events А1, А2,…, Аn, forming a complete system, provided one knows the probabilities thereof and the conditional probabilities of event F relative to each of them PA1 (F )PA2 (F ),..., PAn1 (F ) and so forth, is calculated by the total probability formula:
P(F ) = P( A1)PA1 (F ) + P( A2 )PA2 (F ) + ... + P( An )PAn (F ) =
n
=∑P( Ai )PAi (F ).
i=1
One can obtain a Bayesian formula (a hypotheses formula) by generalizing the product rule so considering that if P(АВ)=Р(А)РА(В)=Р(В)РВ(А), then РА(В)=Р(В)РB(А)/Р(А), or by generalizing for п events, and by using the total probability formula, we have:
PF ( Ai ) = nP( Ai )PAi (F ) .
∑P( A j )PA j (F ) j =1
Bayesian formula is often referred to as formula of hypotheses because it can be used to re-estimate the probability of hypotheses as to the occurrence of the events A1, A2 , A3 ,...An after it became known that event F had occurred.
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Random quantities, distribution and characteristics thereof
We will hold random quantity as a variable that is capable of taking any such value as the case may require. Random quantity is any variable x having values that form a set of elementary events or denote points in a sample space.
The respective distribution of probabilities is called probability of the random quantity x. All random quantities are intrinsically divisible into discrete and continuous quantities.
A discrete random quantity has the set of its values finite or countable (number of items in storage, number of fuel elements, number of containers with NM, etc.).
The function р(х) that relates the value of a random quantity to the corresponding probabilities is called random quantity law. Distribution law can be presented in a tabulated or graphic form. However, it is rather difficult to get a complete idea of random quantity from its law. So certain constants of its law are used. These are the so-called moments of various orders. They provide as complete characterization of random quantity as possible. The most important moments are mathematical expectation (firstorder moment about zero) and variance (second-order moment about mean) [3].
Mathematical expectation of discrete random quantity
Mathematical expectation of a discrete random quantity is the total of the products of all its values by corresponding probabilities:
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M ( X ) = x1 p1 + ... + xi pi + ... = ∑xi pi . |
(4.1) |
i =1
Mathematical expectation of a random quantity is a constant showing what value of the random quantity should be expected (on the average) during tests or observations.
We shall discuss the most important properties of mathematical expectation.
1.Mathematical expectation of a constant is equal to this constant: М(k) = k, where k = const.
2.Mathematical expectation of the sum (the difference) of a finite number of random quantities is equal to the sum (the difference) of the mathematical expectations thereof:
М(Х ± Y) = М(Х) ± М(Y).
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3. Mathematical expectation of the product of a finite number of independent random quantities is equal to the product of the mathematical expectations thereof:
М(ХY) = М(Х)М(Y).
4. Reducing (increasing) all values of the random quantity Х by K = const gives the mathematical expectation a decrease (an increase) by K:
М(Х – K ) = М(Х) – K .
Variance of random quantity
The most commonly encountered measure (characteristic) of a deviation or spreading (scatter) of a random quantity is variance and the root-mean- square deviation (RMSD) obtained from this.
As the scatter (deviation) of a random quantity’s values from the mathematical expectation cannot be evidently characterized by linear differences, the adopted practice is to estimate deviation squares.
Therefore we will call the variance of the random quantity Х the mathematical expectation for the square of its value deviation from the mathematical expectation:
D( X ) = y2 ( X ) = M [X − M ( X )]2 , |
(4.2) |
for discrete random quantities we have: |
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D(X ) = ∑(xi − a )2 pi , |
(4.3) |
where а = М(Х).
The RMSD is equal to:
y( X ) = |
D(x) |
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(4.4) |
The variance of a random quantity is a constant. If the variance is small, all numbers of the sum are also small (4.3). So if хi exists that deviates widely from а, these are unlikely. If the variance is great, this indicates to a big probability of the random quantity values being very remote from а.
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The average root-mean-square deviation σ ( X ) is convenient in that, while characterizing the deviation (spreading) degree, it has the dimension of the quantity measured.
Let us look into the basic properties of variance.
1.The variance of a constant is equal to zero: D(k) = 0; k = const.
2.D(kx) = k 2 D(x) , where k = const.
3.The variance of a random quantity is equal to the mathematical expectation of its square with no mathematical expectation square:
D( X ) = M ( X 2 ) − M 2 ( X ) . |
(4.5) |
4. The variance of a sum (a difference) of independent random quantities is equal to the sum of the variances thereof:
D( X ± Y ) = D( X ) + D(Y ) . |
(4.6) |
It follows from 4 that the average standard deviation of the sum of independent random quantities is equal to the root of the sum of squares for the RMS deviations thereof:
σ (x) = |
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∑σ i2 . |
(4.7) |
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i =1 |
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Consider n similarly distributed random quantities:
if x1, x2 , x3 ,..., xn are similarly distributed random quantities, the mathematical expectation for each of these being equal to а, the mathematical expectation for the sum thereof is equal to na, and the mathematical expectation of the arithmetic mean is equal to а:
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x1 + x2 + ... + xn |
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+ ... + xn ) = |
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na = a ; |
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if x1, x2 , x3 ,..., xn |
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quantities, the variance for each of these being σ 2 , the variance of the sum
thereof is nσ 2 , and the variance of the arithmetic mean is σ 2 : n
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