The characterization of probability distribution is important before its application to any real world phenomena to confirm whether the given probability distribution is suitable for the specific data set or not. There are several characterizing functions associated with a probability distribution that uniquely define it.

We assume that the distribution of random variable called “Z” follows Power function distribution (PFD) with following cumulative density function

Then

The application of this distribution was discoursed by Meniconi and Barry ^{2}, Basu ^{3}, Govindarajulu ^{4} and Dallas ^{5}. Many authors have studied the characterization of different distributions; see Dallas ^{6}, Deheuvels ^{7}, Gupta ^{8}, Nagaraja ^{9}, Rao et al. ^{10} and Huang et al. ^{11}.

Su and Huang

Where g may be observed as continuous function and (a,b) is the support of

Gupta and Kirmani ^{13} consider the case a=0 and b=∞ and used the ratio of ^{14} showed that, for a continuous function g and a differential function h, the pdf “f” satisfies the differential equation. Unnikrishnan and Sudheesh^{16}, Huang et al. ^{18}, Ahsanullah et al. ^{19} and references therein. Imen et al. ^{20} discussed the characterization of Exponential q-distribution. ^{21, 22 and 23} discussed the properties and characterizations of the J shaped distributions. Zaka et al. ^{24} introduced the exponentiated generalized class of Power function distribution.

This study discusses a novel characterization of PFD established on mean inactivity times (MIT), mean residual function (MRF), conditional moments, conditional variance, doubly truncated mean (DTM), incomplete moments and reverse hazard function (RHF). The paper is organized as follows. We characterized the PFD by MIT, MRF and doubly truncated mean in section-2. Under Section -3, we introduce characterization based on different types of moments. In Section 4, we obtain the PFD from RHF. The concluding remarks are provided in Section 5.

We may write the pdf and cdf for the PFD

Also let

Where M_{z}(t): Mean Inactivity Times for PFD.

Proof:

Necessary part:

From (1), by taking β=1.

Since

Since

Also

Sufficient Part:

Now

Using integration by parts

Equating (2) and (3), we get

Differentiating “t” on both sides

Apply Integral on both sides

ln (

Therefore

and f(t)=

so the random variable “z” has PFD, “f(z) =

We may write the pdf and cdf for the PFD

Also let

Where M (t): Mean Residual Function.

Proof:

Necessary part:

Since

Therefore, (4) becomes

Sufficient Part:

Now

Using integration by parts

By equating (5) and (6)

Since

Differentiating “t” on both sides

Therefore,

Therefore, it is proved that the random variable “z” has PFD, f(z) =

We may write the pdf and cdf for the PFD

And let

Proof:

Necessary Condition:

Since

Sufficient Condition:

Also

By equating (7) and (8), we get

Differentiating with respect to ‘w’ on both sides

By integrating on both sides

ln (

Therefore

f(w)=

We may write the pdf and cdf for the PFD

And let

Proof:

Necessary part:

Since

Since

Also

Therefore (9) becomes

Sufficient Part:

Now

Using integration by parts

Equating (10) and (11), we get

Since

Differentiating “t” on both sides

ln (

Therefore

and f(t)=

Therefore, it is proved that the “z” has PFD f(z) =

We may write the pdf and cdf for the PFD

And let

Proof:

Necessary part:

Since

Sufficient Part:

Now

Using integration by parts

Equating (12) and (13), we get

Differentiating “t” on both sides

Apply Integral on both sides

ln (

Therefore

and f(t)=

Therefore it is proved that “z” has PFD, “f(z) =

We may write the pdf and cdf for the PFD

And let

Proof:

Necessary part:

Sufficient Part:

Now

Using integration by parts

Equating (14) and (15), we get

Since

Differentiating “t” on both sides

Therefore

and f(t)=

Therefore it is proved that the “z” has PFD, f(z) =

We may assume_{ z}(t)>0 , r_{w}(t)=a r_{z}(t) with a>0 iff

Where U(t)=

“r_{z}(t)” and “r_{w}(t)” are the reverse hazard rates. Also follows the PFD.

Proof:

Necessary Part:

Since r_{w}(t)=a r_{z}(t)

Since

Sufficient Part:

Taking derivative on both sides

Also (16) becomes

The objective of the study is to acquire the characterization of PFD based on mean inactivity times (MIT), mean residual function (MRF), conditional moments, conditional variance, doubly truncated mean (DTM), incomplete moments and reverse hazard function (RHF). We hope that this study will be useful for the statisticians in various fields of studies.