1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

2.3. Independence. When desiring to consider the completestochastic characteristic of events A 1 , A 2 , ..., A n , we will need to knowthe probabilities of every possible setP(C 1 , C 2 , ..., C n )where each C i can take two values, A i and Ai.It is not difficult tocalculate that 2 n probabilities are needed. This number increases veryrapidly with n and the pertinent possibilities of any experiment becomeinsufficient. We expect such stochastic models to be applicable only ifthat difficulty is somehow overcome and the main part is played hereby the concept of independence.Definition. Two events, A and B, are independent if the conditionalP(A/B) and unconditional probabilities coincide:P( AB)P( A / B) = P( A) or P( AB) P( A) P( B).P( B )= =For n events A 1 , A 2 , ..., A n independence is defined by equalityP(C 1 C 2 , ..., C n ) = P(C 1 ) P(C 2 ) ... P(C n ) (2.4)where each C i can take values A i and Ai.Since P( Ai) = 1 – P(A i ), theprobabilities for independent events can be given by only n valuesP(A 1 ), P(A 2 ), ..., P(A n ).Independent events do exist; they are realized in experiments carriedout independently one from another (in the usual physical meaning). Atextbook on the theory of probability should show the reader how thecorresponding space of elementary events is constructed here, but thisbooklet is not a textbook. I have provided a sufficiently detailedexposition of the most essential notions of that theory so as to showhow it is done, briefly and conveniently (one concept, one axiom, onedefinition) in the set-theoretic language. The further development isalso offered briefly and conveniently, but from the textbook style I amturning to the style of a summary.2.4. Random variables. Definition. A random variable is a functiondefined on a set of elementary events. They are usually denoted byGreek letters ξ, η, ζ etc. When desiring to include the argumentω ⊂ Ω , we write ξ(ω), η(ω), ζ(ω) etc.A set of possible values a 1 , a 2 , ..., a n , ... of events, all of themdifferent,{ω: ω ⊂ Ω , ξ(ω) = a i } = {ξ= a i }is connected with each random variable ξ = ξ(ω), as well asprobabilities∑P{ξ = a } = P(ω) = p , ω:ξ(ω) = a .i i i19