Page 12 - DMTH404_STATISTICS

Statistics

Notes In Experiment 4, the person coming out of the polling centre may give us the name of the
candidate for whom helshe voted, or may refuse to disclose hisher choice. If there are 5 candidates
C , C , C . C and C , seeking election, then there are six possible outcomes, five corresponding
1 2 3 4 5
to the five candidates and the sixth one corresponding to the refusal R of the interviewed person
to disclose hisher choice. The set of all possible outcomes is thus, {C , C , C * C , C , R}.
1 2 3 4 5
Note that here we have ignored certain possibilities, like the possibility of the person not voting
at all or voting in such a manner that hisher ballot paper becomes invalid.
Experiment 5 is comparatively simple, if we agree that it is possible to classify each item as
Good (G) or Bad (B) without error. Then R = {GGG, GGB, GBG, BGG, BBG, BGB, GBB, BBB} where.
for example, GBG denotes the outcome when the first and third units are good and the second
one is bad.
The situation in Experiment 6 is a little more complicated. To test the efficacy of the vaccine, we
will have to look at the number of vaccinated persons who were affected (x) q the number of
non-vaccinated ones who were affected (y). Here x can be any integer between 0 and 30 and y can
be any integer between 0 and 20. The set  of all possible outcomes is
 = {(x,y) | x = 0, 1, ..., 30, y = 0, l, 2,..., 20}.
This specification of  is valid only if we assume that we are able to observe all the 50 persons
for the entire period of six months. In particular, we assume that none of them becomes
untraceable because of hisher leaving the town or because of hisher death due to some other
cause.
In the illustrations discussed so far, do you notice that the number of points in  is finite in each
case? It is 2 for Experiment 3,6 for ExIjeriment 4,31 x 21 = 651 for Experiment 6. But this is not
always true.
Consider, for example, Experiments 9 and 10. The number of accidents along the Bombay-
Bangalore highway during the month of observation can be zero, one, two, . . . or some other
positive integer. Similarly, the number of a-particles emitted by the radio-active substance can
be any positive integer. Can we say that the number of accidents or a-particles would not exceed
a specified limit? No. Because of this, and also in order to simplify our mathematics, we usually
postulate that in both these examples the set of all possible outcomes is R = {0, 1,2, ...}, i.e., it is the
set of all non-negative integers.

We are now in a position to introduce certain terms in a formal manner.
Definition 2 : The set  of all possible outcomes of an experiment E is called the sample space of
the experiment. Each individual outcome of E is called a point, a sample point or an element
of .
You would also notice that in every experiment th,at was discussed, we made certain assumptions
like the coin not being able to stand on its edge or not rolling away, all the fifty persons being
available for the entire period of six months for observation, etc. Such assumptions are necessary
to simplify our problems as well as our mathematics.
In all the examples discussed so far, the sample space is either a finite set, i.e., a set containing a
finite number of points or is an infinite set whose elements can be arranged in an unending
sequence, i.e., has a countable infinity of elements. We have a special name for such spaces.
Definition 3 : A sample space containing a finite number of points or a countable infinity of
points is called a discrete sample space.

In this block we shall be concerned only with discrete sample spaces. However, there are mhy
situations where we have to deal with sample spaces which are not discrete. For example,
consider the age of a person. Although there are limitations to the accuracy with which we can

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