Lecture 1/ 2017-2018/ Random Experiment

Probability
Random Experiment
It is an experiment that its outcomes can not be predicted, and each its outcome has the same chance(probability).
Example
Tossing one single fair coin one time.
Sample space=?={H,T}
Let X be number of appearance the head, that P(H)=1/2 then P(T)=1/2
Now let X=1 for appearance the head, and let X=0 for without appearance the head, e.g. P(H)=P(X=1)=1/2 then
P(T)=P(X=0)=1/2 and P(X=1)+P(X=0)=1
Boolen Algebra (Algebra)
It is a set of all possible subsets of the sample space ?,denoted as A, which are called events. Therefore this set has the following properties:
1) ??A,
2) A?A?A^c?A,
3) A_1,A_2?A?A_1?A_2?A.
Sigma-algebra(?-algebra)
Let A be non empty set. Then a collection of all subsets of A, denoted A,N,??,..
It is called ?-algebra iff it has the following properties:
1) ??A,
2) A?A?A^c?A,
Finite Sample Space with Equally Points
It is a sample space with finite number of outcomes as N, then it is called
finite sample space with equally points if the probability of each its outcome is 1/N . Let P[.] be a probability function, where P: A?[0,1] and A is event space satisfies the following conditions:
1) P[{w_1 }]=P[{w_2 }]=?=P[{w_N }]
2) A?A?P[A]=N(A)/N, where N(A)= number of elements that A contains, and ?={w_1,w_2,…,w_N}. Then it is readily checked that the set function P[•] satisfies the three axioms and hence is a probability function.