Page 59 - DMTH404_STATISTICS

Unit 4: Random Variable

Notes
4 C 4
P(X = 3, i.e., 3R marbles are drawn)  3 
6
C 20
3

Notes In the event of white balls being greater than 2, the possible values of the
random variable would have been 0, 1, 2 and 3.

4.2.2 Cumulative Probability Function or Distribution Function

This concept is similar to the concept of cumulative frequency. The distribution function is
denoted by F(x).
For a discrete random variable X, the distribution function or the cumulative probability function
is given by F(x) = P(X £ x).
If X is a random variable that can take values, say 0, 1, 2, ......, then
F(1) = P(X = 0) + P(X =1), F(2) = P(X = 0) + P(X =1) +P(X = 2), etc.
Similarly, if X is a continuous random variable, the distribution function or cumulative
probability density function is given by
x
F   x  P X £  x  -¥ ò p ( )dX
X

4.3 Summary

 A random variable X is a real valued function of the elements of sample space S, i.e.,
different values of the random variable are obtained by associating a real number with
each element of the sample space. A random variable is also known as a stochastic or
chance variable.

Mathematically, we can write X = F(e), where e ÎS and X is a real number. We can note here
that the domain of this function is the set S and the range is a set or subset of real numbers.

 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.
 When X is a continuous random variable, there are an infinite number of points in the
sample space and thus, the probability that X takes a particular value is always defined to
be zero even though the event is not regarded as impossible. Hence, we always measure
the probability of a continuous random variable lying in an interval.
 The concept of a probability distribution is not new. In fact it is another way of representing
a frequency distribution. Using statistical definition, we can treat the relative frequencies
of various values of the random variable as the probabilities.

4.4 Keywords

Random variable: A random variable X is a real valued function of the elements of sample space
S, i.e., different values of the random variable are obtained by associating a real number with
each element of the sample space.

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