Mathematical Preliminaries 5 X:2→R” (1.14) is an n-dimensional random variable or n-dimensional random vector. Definition The probability distribution of the random variable X is a function Px:B[0,1] such that Px(b)=P(X∈B) (1.15) be B The probability distribution of the random variable is a probabilistic measure. Definition Let us consider the following probability space (,B,Px).The function Fx:gr→[0，1]defined as F(x)=Px《-∞，xl (1.16) is called the cumulative distribution function of the variable X. Definition The function f:,has the following properties: (1)there holds almost everywhere (in each point of the cumulative distribution function differentiability): dF(x)=f(x) (1.17) dx (2) f(x)20 (1.18) (3) Jrcod=1 (1.19) (4)for any Borelian set beB the integral ff(x)dx=P(Xeb)is a probability density function(PDF)of the variable X. Definition Let us consider the random variable X:defined on the probabilistic space (F,P).The expected value of the random variable X is defined as EX]=∫X(o)dP(o) (1.20)Mathematical Preliminaries 5 n X :Ω → ℜ (1.14) is an n-dimensional random variable or n-dimensional random vector. Definition The probability distribution of the random variable X is a function ] P : B →[0,1 X such that P (b) P(X B) X b B ∀ = ∈ ∈ (1.15) The probability distribution of the random variable is a probabilistic measure. Definition Let us consider the following probability space ( ) B PX ℜ, , . The function :ℜ →[0,1] FX defined as F x P [ ] ( ) x X ( ) = − ∞, (1.16) is called the cumulative distribution function of the variable X. Definition The function ℜ → ℜ+ f : has the following properties: (1) there holds almost everywhere (in each point of the cumulative distribution function differentiability): ( ) ( ) f x dx dF x = (1.17) (2) f (x) ≥ 0 (1.18) (3) ∫ +∞ −∞ f (x)dx = 1 (1.19) (4) for any Borelian set b ∈ B the integral ( ) ∫ = ∈ b f ( is a probability x)dx P X b density function (PDF) of the variable X. Definition Let us consider the random variable X :Ω → ℜ defined on the probabilistic space ( ) Ω, F, P . The expected value of the random variable X is defined as ∫ +∞ −∞ E[X ] = X (ω)dP(ω) (1.20)