Page 17 - DMTH404_STATISTICS

Unit 1: Sample Space of A Random Experiment

The union and intersection of two sets can be utilised to define union and intersection of three or Notes
more sets.
So, if A , A , . . . , A are n events, then we define
1 2 n
n
 A j = {  |   A for every j = l, ..., n }.
j 1 j

and
n
 A j = {  |   A for at least one j = 1, ..., n }.

j 1 j
n
Note that the occurrence of  A j corresponds to the simultaneous occurrence of all the n events

j 1
n
and the occurrence of  A j corresponds to that of at least one of the n events A , . . . , A . We can
j 1 1 n

similarly define the union and intersection of an infinite number of events, A , A , . . . ., A , . . . .
1 2 n
Another set operation with which you are familiar is a combination of complementation and
C
intersection. Let A and B be two sets. Then the set AB is usually called the difference of A and
B and is denotedby A – B. It consists of all points which belong to A but not to B.
Thus, in Example 4,

B – B = { (6, 1), (6, 2), (6, 3), (6, 4) }
1 2
and
B – B = { (5,6) }
2 1
C
In this notation, A is the set  – A. You can see the Venn diagram for A – B in Fig. 4.
Now, suppose A , A and A are three arbitmy events. What does the occurrence of A  A  A c
c
1 2 3 1 2 3
signify?
This event occurs iff only A out of A , A and A occurs, that is, iff Al occurs but neither A nor
1 1 2 3 2
A occur.
3
If you have followed this, you should be able to do this exercise quite easily.

Task If A , A and A are three arbitrary events, what does the occurrence of the
1 2 3
following events signify?
(a) E = A A A
1 1 2 3
c
(b) E = A  A  A c
c
2 1 2 3
c
(c) E = (A  A  A )  (A  A  A )  (A  A  A )
c
c
3 1 2 3 1 3 2 2 3 1
(d) E  E
1 3

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