PDF of the Random Variable when its Distribution Function Changes after the Change Points

This paper derives the probability density function of the random variable X when the random variable X changes from one distribution to another after the change points. Illustrations are provided. Mathematics Subject Classification: 60A05, 60A99, 60E05


Preliminaries
According to the probability theory the cumulative distribution function (CDF) describes the probability that a real valued random variable X with a given probability distribution will be found at a value less than or equal to x.In the case of a continuous distribution, it gives the area under the probability density function (PDF) from െ∞ to x.

Definition 1.1
The cumulative distribution function of a real-valued random variable X is the function given by

‫ܨ‬ ሺ‫ݔ‬ሻ ൌ ܲሺܺ ‫ݔ‬ሻ
where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x.

Definition 1.2
The probability that X lies in the interval (a, b), where a < b, is therefore The CDF of the random variable X can be expressed as the integral of its PDF as follows.
The complementary CDF of the random variable X is defined and denoted as follows.

Remark 1.7
If the events are independent, then Obviously, we can find that from the definition 1.1, the CDF is very much connected with the event that the random variable X < x.

A distribution having One Change Point
Suppose the random variable ܺ follows the PDF ݄ ଵ ሺ‫ݔ‬ሻ with the CDF ‫ܪ‬ ଵ ሺ‫ݔ‬ሻ when 0 ൏ ‫ݔ‬ ൏ ߬ ଵ and it follows the PDF ݄ ଶ ሺ‫ݔ‬ሻ with the CDF ‫ܪ‬ ଶ ሺ‫ݔ‬ሻ when ߬ ଵ ൏ ‫ݔ‬ ൏ ∞.
(Since the change of distribution occurs at ଵ , it is called as the change point) Using Remarks 1.7, 1.8 and definition 1.6 for any two events, the CDF of X can be given as follows.
Simplifying (2.1) we get the following form By differentiating (2.2), we get the PDF ݄ሺ‫ݔ‬ሻ of ܺ.It is given by
Using Remarks 1.7, 1.8 and definition 1.6 for any three events, the CDF of X can be given as follows.
Simplifying (2.4), we get the following form We verify that (2.6) is a PDF.

A distribution having 'n-1' Change Points
Suppose the random variable ܺ happens to change its distribution 'n-1' times, with change points ߬ ଵ , ߬ ଶ,⋯ ߬ ିଵ then its PDF can be generalized as follows.

Illustration a) A distribution having one change point
3.1 Suppose the random variable ܺ~expሺߠ ଵ ሻ before τ and X~expሺߠ ଶ ሻ after ߬ then using 2.3 its pdf can be given by This pdf has been used in [3].

C. D. Nanda Kumar and S. Srinivasan
3.3 its pdf can be given by This pdf has been used in [2] and [3].
3.4 Suppose the random variable ܺ~expሺߠሻ before τ and X~gamma 2ሺkሻ after ߬ then using 2.3 its pdf can be given by This pdf has been used in [1].

Future Work
The future work of this paper is deriving the pdf of the random variable x by treating the change point as the random variable.