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Unit 4: Random Variable

Solution. Notes

The sample space of the experiment can be written as
S = {(H,H,H), (H,H,T), (H,T,H), (T,H,H), (H,T,T), (T,H,T), (T,T,H), (T,T,T)}
We note that the first element of the sample space denotes 3 heads, therefore, the corresponding
value of the random variable will be 3. Similarly, the value of the random variable corresponding
to each of the second, third and fourth element will be 2 and it will be 1 for each of the fifth, sixth
and seventh element and 0 for the last element. Thus, the random variable X, defined above can
take four possible values, i.e., 0, 1, 2 and 3.

It may be pointed out here that it is possible to define another random variable on the above
sample space.

4.2 Probability Distribution of a Random Variable

Given any random variable, corresponding to a sample space, it is possible to associate
probabilities to each of its possible values. For example, in the toss of 3 coins, assuming that they
are unbiased, the probabilities of various values of the random variable X, defined in example
1 above, can be written as :
1 3 3 1
P
X   0  , X   1  , X   2  and P X   3  .
P
P
8 8 8 8
The set of all possible values of the random variable X along with their respective probabilities
is termed as Probability Distribution of X. The probability distribution of X, defined in example
1 above, can be written in a tabular form as given below :
X : 0 1 2 3 Total
1 3 3 1
p   :X 1
8 8 8 8
Note that the total probability is equal to unity.
In general, the set of n possible values of a random variable X, i.e., {X , X , ...... X } along with
1 2 n
n
their respective probabilities p(X ), p(X ), ...... p(X ), where å p   1X  , is called a probability
1 2 n i
i 1
distribution of X. The expression p(X) is called the probability function of X.
4.2.1 Discrete and Continuous Probability Distributions

Like any other variable, a random variable X can be discrete or continuous. If X can take only
finite or countably infinite set of values, it is termed as a discrete random variable. On the other
hand, if X can take an uncountable set of infinite values, it is called a continuous random
variable.
The random variable defined in example 1 is a discrete random variable. However, if X denotes
the measurement of heights of persons or the time interval of arrival of a specified number of
calls at a telephone desk, etc., it would be termed as a continuous random variable.
The distribution of a discrete random variable is called the Discrete Probability Distribution
and the corresponding probability function p(X) is called a Probability Mass Function. In order
that any discrete function p(X) may serve as probability function of a discrete random variable
X, the following conditions must be satisfied:

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