Page 94 - DMTH404_STATISTICS

Statistics

Notes Sample Space

It is the set of all possible outcomes of a random experiment. Each element of the set is called a
sample point or a simple event or an elementary event. The sample space of a random experiment
is denoted by S and its element are denoted by e , where i = 1, 2, ...... n. Thus, a sample space
i
having n elements can be written as :
S = {e , e , ......, e }.
1 2 n
If a random experiment consists of rolling a six faced die, the corresponding sample space
consists of 6 elementary events. Thus, S = {1, 2, 3, 4, 5, 6}.
Similarly, in the toss of a coin S = {H, T}.
The elements of S can either be single elements or ordered pairs. For example, if two coins are
tossed, each element of the sample space would consist of the set of ordered pairs, as shown
below :
S = {(H, H), (H, T), (T, H), (T, T)}

Finite and Infinite Sample Space

A sample space consisting of finite number of elements is called a finite sample space, while if the
number of elements is infinite, it is called an infinite sample space. The sample spaces discussed so
far are examples of finite sample spaces. As an example of infinite sample space, consider repeated
toss of a coin till a head appears. Various elements of the sample space would be :
S = {(H), (T, H), (T, T, H), ...... }.

Discrete and Continuous Sample Space

A discrete sample space consists of finite or countably infinite number of elements. The sample
spaces, discussed so far, are some examples of discrete sample spaces. Contrary to this, a continuous
sample space consists of an uncountable number of elements. This type of sample space is
obtained when the result of an experiment is a measurement on continuous scale like
measurements of weight, height, area, volume, time, etc.

Event

An event is any subset of a sample space. In the experiment of roll of a die, the sample space is
S = {1, 2, 3, 4, 5, 6}. It is possible to define various events on this sample space, as shown below :
Let A be the event that an odd number appears on the die. Then A = {1, 3, 5} is a subset of S.
Further, let B be the event of getting a number greater than 4. Then B = {5, 6} is another subset of
S. Similarly, if C denotes an event of getting a number 3 on the die, then C = {3}.

It should be noted here that the events A and B are composite while C is a simple or elementary
event.

Occurrence of an Event

An event is said to have occurred whenever the outcome of the experiment is an element of its
set. For example, if we throw a die and obtain 5, then both the events A and B, defined above, are
said to have occurred.
It should be noted here that the sample space is certain to occur since the outcome of the
experiment must always be one of its elements.

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