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2.4 Probabilistic and statistical definitions and notations
Many uncertainty management tools are based on statistical and probabilistic concepts. So before going further, the following is an introduction to the basics of the probability theory.
The reader can refer to Walter Appel’s book [App13]. The following definitions are extracted from [Sap06] and [Bea04].
2.4.1 Probability space
A random experiment is an experiment for which the results can not be predicted beforehand.
The set of all possible outcomes of the experiment is called the sample space and is generally denoted Ω. An element of Ω is an elementary event denoted ω, withω ∈Ω. A random event is defined as a set of outcomes ω.
The intersection of two eventsAandBis denotedA∩B. Respectively, their union is denoted A∪B.
The set of possible events associated to the sample space Ω is called an event space. This space forms a class F of part of Ω also called σ-algebra. It is defined by the three following axioms:
1. if A∈ F then ¯A∈ F (with ¯A complement of A), 2. if A1,A2, ...,An ∈ F thenSn
i=1Ai ∈ F, 3. F is non-empty then Ω∈ F.
The pair (Ω, F) is called probabilisable space, with F aσ-algebra associated to Ω.
Each event is equipped with probabilities. They correspond to positive numbers which lie between 0 and 1. The probability of an eventE is denotedP(E). This probability is defined by the axiom of Kolmogorov given by definition 2.4.1.
Definition 2.4.1. [Kolmogorov axiom] A probability is an application P withP: F 7→[0,1]
such as:
P(Ω) = 1,
P(A)≥0, ∀A∈ F,
P(A∪B) =P(A) +P(B), ∀A, B∈ F such thatA∩B =∅.
From the Kolmogorov axiom, we have the properties of aprobability measure. P is a proba-bility measure if:
P(∅) = 0,
P( ¯A) = 1−P(A),
P(A)≤P(B), if A⊂B,
P(A∪B) =P(A) +P(B)−P(A∩B).
The triplet (Ω, F,P) is called a probability space.
2.4.2 Random variable
A random variable (or stochastic variable) X is a variable whose values are outcomes from a random process. More precisely, a real random variable is an application from the probability space (Ω, F, P) to the space of real number R. There are two types of random variables:
continuous variable and discrete variable. A random variable X is continuous if its domain of definition contains a real interval. Inversely, X is discrete if it can take a finite or a countable infinite set of discrete values. A random variable is generally associated with two functions:
Cumulative Distribution Function and Probability Density Function.
2.4.2.1 Cumulative Distribution Function
The Cumulative Distribution Function (CDF) of a random variableX is the applicationFfrom Rto [0,1] given by:
F(x) =P(X≤x).
It is an increasing monotoneous function which describes the probability for the random variable Xto be less or equal to a valuex. It is defined for both discrete and continuous random variables.
An example of such function is exhibited in figure 2.8.
By using the CDF, it is possible to determine the probability of x lying in a range [a, b] with a≤b:
P(a≤X≤b) =F(b)−F(a).
Figure 2.8: Cumulative Distribution Function of a continuous variable 2.4.2.2 Probability Density Function
The Probability Density Function (PDF) is defined only for continuous random variables. In-deed, a continuous random variable X has a density function denotedfX and defined by:
P(a≤X≤b) = Z b
a
fX(x)dx.
The PDFfX is a positive function with integral equal to 1:
Z
R
fX(x)dx= 1.
Figure 2.9: Probability Density Function
2.4.3 Statistical measure
A random variableX with a known PDF is characterised by some statistical criteria.
2.4.3.1 Mean
The mean (also called mathematical expectation or expected value) of a random variable X, denoted E(X), is defined by
• E(X) =R
RxfX(x)dx, if X is a continuous variable (a variable with a density function)
• and E(X) =P
ixiP(X=xi), ifX is a discrete variable.
It gives the measure of the central tendency of values taken by X. The mean is also called first moment. It is commonly denoted µX.
2.4.3.2 Variance and Standard-deviation
The variance of a random variableX, denoted V(X) is defined by:
V(X) =E((X−E(X))2) =E(X2)−(E(X))2.
It is a measure of the dispersion of X around its mean valueE(X). The variance is called second moment and commonly denoted σ2 =. From the definition of the variance, we can define the standard-deviation σ. It is the square root of the variance, meaningσX =p
V(x).
2.4.3.3 Other moments
The moment of ordern is defined by:
mn(X) =E(Xn).
In the literature, we can find the notion of centred moment and normalised (or standardised) moment. The centred moment of order nis defined by:
mn(X) =E((X−E(X))n), and the normalised moment of ordern is given by:
mn(X) =E
X−E(X) σ
n .
Let us particularly consider the third and the fourth moments. These two moments can give information about the shape of the distribution of a random variableX.
The third normalised moment is called skewness coefficient. It is the moment of order 3 and its centred expression is given by E (X−E(X))3
. Its normalised expression corresponds to the centred moment of order 3 normalised by the cube of the standard-deviation σ and is defined by:
The skewness coefficient gives information about the symmetry of the density function. A dis-tribution with a negative skewness means that the disdis-tribution is asymmetric to the left (with a tail longer to the left than to the right). A distribution with a positive skewness means that the distribution is asymmetric to the right. And a distribution with a skewness equal to zero means that the distribution is symmetric. As an example, the normal density function has a symmetrical distribution then its skewness coefficient is equal to zero.
The fourth normalised moment is called kurtosis coefficient, also called Pearson Kurtosis coefficient. Its centred definition is given byE((X−E(X))4) and its normalised one is defined by:
The kurtosis coefficient gives information about the ”peaky” aspect of the distribution: it measures the degree of the concentration of the observations in the distribution tail. The higher the value of the kurtosis coefficient, the more peaky the distribution. Kurtosis values are used to be compared to the kurtosis value of the Gaussian distribution which is equal to 3. A new expression of the kurtosis coefficient is given taking the Gaussian distribution as reference. It is called the excess Kurtosis: where n is the size of the sample. A distribution with a positive excess Kurtosis coefficient is peaky: it rises up and falls abruptly. It is called leptokurtic distribution. One with a negative excess Kurtosis coefficient is flattened and is called platykurtic distributions.
2.4.3.4 Covariance
The covariance of two random variablesX andY is given by:
cov(X, Y) =E(XY)−E(X)E(Y) =E
(X−E(X))(Y −E(Y)) .
The covariance is a measure of the strength of thelinear link between two random variables. The variance is a particular case of the covariance: cov(X, X) =E
(X−E(X))(X−E(X))
=V(X).
The covariance coefficient depends on the units of the variablesX andY. It can be meaningless to compare the covariance of variables which are totally different (such as the mass of the fuselage and the sweep angle). To avoid this drawback, one can use the correlation coefficient.
2.4.4 Correlation
2.4.4.1 Correlation coefficient
The correlation coefficient, also called Pearson’s coefficient in the literature, is a normalisation of covariance coefficient by the product of the standard deviation of the variables and is independent of the variable’s units. It gives information on the existence of a linear relation between two variables. The correlation coefficient betweenX andY is given by:
r(X, Y) = E[(X−µX)(Y −µY)]
σXσY , (2.9)
whereE[(X−µX)(Y −µY)] is nothing but the covariance between X and Y commonly denoted cov(X, Y). The correlation coefficient r(X, Y) is always in [−1,1] and it is equal to:
• 0 if there is no linear relation between the samples;
• 1 if there is a maximal positive linear relation and it is in the range [0,1] if there is a simple positive linear dependence;
• -1 if there is a maximal negative linear relation and it is in the range [-1, 0] if there is a simple negative linear relation.
The correlation coefficient presents some drawbacks:
• It is sensitive to outliers i.e. values which are far from the others and which are considered as exceptions.
• It only gives information on the existence of a linear relation between two variables. Thus, two variablesX and Y which are independent have a correlation coefficient equal to zero.
Note that the converse is wrong. The correlation coefficient can be equal to zero even though there is a ’non linear’ functional relation between the variables. Consider the functionY =X2. Y is completely dependent ofX but their correlation coefficient is equal to zero.
• It is a biased coefficient but this bias disappears when the size of the sample increases.
• Its interpretation is sensitive. A high correlation coefficient between two variables doesn’t necessarily mean that there exists a causal relationship between them. It could just mean that the phenomenon represented by the variables have the same origin. For instance, let us consider the correlation between ”the number of firemen on the fire area” and ”the repayment price demanded by the insurance company”. A study has shown that there is a high correlation between these events but it is obvious that in reality it is not the case.
Although, if we consider the third event ”the fire gravity”, we see that the first two ones have the same cause: the third one.
In the case where the relation between the variables is non linear but monotonic, the corre-lation coefficient can highlight the existence of a recorre-lation but is not able to give information on the strength of the relation. One solution to avoid that problem is to make a transformation (of the variables) in order to obtain a linear relation. Nevertheless, this way of doing presents several drawbacks: the choice of the transformation function is not always obvious. Moreover, the number of transformation increases with the number of relationship to analyse. In this case, other correlation coefficients such as the Spearman’s one are more relevant.