ONT Re: Probability And Statistics
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PAS. Note 2
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| 1.2. Probability Spaces (cont.)
|
| The next question is, what should the collection $A$ be?
| It is quite reasonable to demand that $A$ be closed under
| finite unions and finite intersections of sets in $A$ as
| well as under complementation.
|
| For example, if A and B are two events, then A |_| B occurs if the
| outcome of our experiment is either represented by a point in A or
| a point in B. Clearly, then, if it is going to be meaningful to
| talk about the probabilities that A and B occur, it should also
| be meaningful to talk about the probability that either A or B
| occurs, i.e., that the event A |_| B occurs. Since only sets
| in $A$ will be assigned probabilities, we should require that
| A |_| B is in $A$ whenever A and B are members of $A$.
|
| Now A |^| B occurs if the outcome of our experiment is represented
| by some point that is in both A and B. A similar line of reasoning
| to that used for A |_| B convinces us that we should have A |^| B
| in $A$ whenever A, B are in $A$.
|
| Finally, to say that the event A does not occur is to say that
| the outcome of our experiment is not represented by a point in A,
| so that it must be represented by some point in A^c. It would be
| the height of folly to say that we could talk about the probability
| of A but not of A^c. Thus we shall demand that whenever A is in $A$
| so is A^c.
|
| We have thus arrived at the conclusion that $A$
| should be a nonempty collection of subsets of !W!
| having the following properties:
|
| 1. If A is in $A$ so is A^c.
|
| 2. If A and B are in $A$ so are A |_| B and A |^| B.
|
| An easy induction argument shows that
| if A_1, A_2, ..., A_n are sets in $A$
| then so are:
|
| |_| (i = 1 to n) A_i
|
| and
|
| |^| (i = 1 to n) A_i.
|
| Here we use the shorthand notation:
|
| |_| (i = 1 to n) A_i = A_1 |_| A_2 |_| ... |_| A_n
|
| and
|
| |^| (i = 1 to n) A_i = A_1 |^| A_2 |^| ... |^| A_n.
|
| Also, since A |^| A^c = {} and A |_| A^c = !W!, we see
| that both the empty set {} and the set !W! must be in $A$.
|
| Hoel, Port, Stone, 'Probability Theory', pp. 6-7.
|
| Hoel, P.G., Port, S.C., & Stone, C.J.,
|'Introduction to Probability Theory',
| Houghton Mifflin, Boston, MA, 1971.
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