Propostion 1.3.2. If are random variables, then is a random vector.
Definition 1.3.3. The distribution function F : R → [0, 1] of a random variable X is defined by the rule
1.4. Some features of random variable
Definition 1.4.1. The expectation (mean) of X is
The usual Lebesgue integral.
Definition 1.4.2. The variance of X is .
Proposstion 1.4.3.
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2. Extremal Finite Set Theory
Definition 2.1. A finite set including n elements is called n-set.
Definition 2.2. Let n-set S. A family ( : the set of all subsets of S) is called antichain if no set of F is contained in another.
Definition 2.3. Given the set S. Let be family of pairs of subsets of S. Then, F is called (k, l)-system for all i and for all .
3. The entropy function
Definition 3.1. Let X be a random variable taking values in some range S, and let px denote the probability that the value of X is x. The binary entropy of X, denoted by H(X) is defined by
Propositon 3.2. Let be a random variable taking values in the set , where each of the coordinates of X is a random variable taking values in . Then
CHAPTER 2: PROBLEMS APPLYING THE PROBABILITY METHOD IN EXTREMAL FINITE SET THEORY
1.. Sloving some problems applyong the probability method in extremal finite set theory
The problems in finite extreme set theory usually take the form: Let S be a set ( a family set ), satisfy the given condition. Find or . When does the equal sign occur?
Let’s consider a simple exercise.
-Problem a. Let F be a family of subset of a set of n-elements X. Suppose, for any two elements of F also have a non-empty intersection. What is the maximum possible value of F?
-Problem a’. Let F be a family of subsets of a set of n-elements X. Suppose, for any A and B of F have . What is the maximum possible value of F?
-Problem b.. Let F be a family of subset of a set of n-elemets X. Suppose, for any A and B of F have and What is the maximum posible value of F?
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