The convexity and monotonicity property of the mapping H in the case when x and y are two discrete random variables taking values (x_{i}) and (y_{i}), respectively, with probabilities (p_{i}), (i=1,ldots,n), was proven in [7].
Supposing that there are two discrete random variables X1 and X2, the corresponding probability function is: left{ begin{gathered} { Pr }{ X_{1} = x_{1i} } = p_{1i}, quad 1 le i le k_{1} hfill { Pr }{ X_{2} = x_{2i} } = p_{2i}, quad 1 le i le k_{2} hfill end{gathered} right.
Conditional Entropy If X and Y are two discrete random variables; P x, y) and P y|x) are joint and conditional probability distributions respectively, then the conditional entropy associated with these distributions is defined as begin{aligned} H(Y|X)=-sum limits _{xin X}sum limits _{yin Y} P x,y log _2 P y|x).
The MI is a measure of the statistical dependence between random variables[15] and is defined for two discrete random variables X and Y as I ( X ; Y i ) = ∑ x ∈ X, y ∈ Y i p ( x, y ) log 2 p ( x, y ) p ( x ) p ( y ).
Let A X, A Y be operators associated with two discrete random variables X and Y for f, g ∈ K. Suppose that α and β are two real numbers, then we easily get the following linear property of the operator in (4): A X ( α f + β g ) = α A X ( f ) + β A X ( g ).
With PF, continuous distribution could be approximated by discrete random measures.
where denotes discrete entropy, and are independent discrete random variables.
Therefore we may assume that N is a discrete random variable.
