# On the Uncertainty Properties of the Conditional Distribution of the Past Life Time

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## Abstract

**:**

## 1. Introduction

## 2. The Past Life-Time Uncertainty in Coherent Systems

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Example 1.**

- (a)
- Let X be uniformly distributed in $[0,1].$ It holds that$$E[log{f}_{t}\left({F}_{t}^{-1}\left({U}_{i:n}\right)\right)]=-log\left(t\right),$$$$\overline{H}\left({T}_{t}\right)=-0.05757+log\left(t\right).$$It is seen that the entropy of ${T}_{t}$ is an increasing function of time $t.$ We note that the uniform distribution has the DRHR property, and therefore, $\overline{H}\left({T}_{t}\right)$ is an increasing function of time t, as we expected based on Theorem 1.
- (b)
- Let us assume that X follows the cdf$$F\left(x\right)={e}^{-{x}^{-k}},\phantom{\rule{4pt}{0ex}}x>0,\phantom{\rule{4pt}{0ex}}k>0.$$One can see that$$E[log{f}_{t}\left({F}_{t}^{-1}\left({U}_{i:n}\right)\right)]=log\left(k\right)+E[log{U}_{i:n}]+\frac{k+1}{k}E\left[log\left({t}^{-k}-log\left({U}_{i:n}\right)\right)\right],$$for all $i=1,2,3,4.$ Upon recalling (9), we obtain$$\overline{H}\left({T}_{t}\right)=1.0257-log\left(k\right)-\frac{k+1}{k}\sum _{i=1}^{n}{p}_{i}E\left[log\left({t}^{-k}-log\left({U}_{i:n}\right)\right)\right],$$for all $t>0.$ For several choices of k, we have shown the exact value of $\overline{H}\left({T}_{t}\right)$ with respect to time t in Figure 1. It is obvious that $\overline{H}\left({T}_{t}\right)$ is an increasing function of time t for all $k>0$ since X is DRHR, as can follow from Theorem 1.

**Lemma 1.**

**Theorem 3.**

**Proof.**

**Corollary 1.**

## 3. Bounds for the Past Entropy

**Theorem 4.**

**Proof.**

**Example 2.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**(i)**- if $H\left({V}_{1}\right)\ge H\left({V}_{2}\right)$ and ${f}_{t}\left({F}_{t}^{-1}\left(u\right)\right)$ increases in u for all $t>0,$ then $\overline{H}\left({T}_{1,t}\right)\ge \overline{H}\left({T}_{2,t}\right)$.
**(ii)**- if $H\left({V}_{1}\right)\le H\left({V}_{2}\right)$ and ${f}_{t}\left({F}_{t}^{-1}\left(u\right)\right)$ decreases in u for all $t>0,$ then $\overline{H}\left({T}_{1,t}\right)\le \overline{H}\left({T}_{2,t}\right)$.

**Proof.**

**Example 3.**

**Corollary 2.**

**(i)**- if ${f}_{t}\left({F}_{t}^{-1}\left(u\right)\right)$ increases in u for all $t>0,$ then $\overline{H}\left({T}_{t}\right)\ge \overline{H}\left({T}_{t}^{D}\right)$.
**(ii)**- if ${f}_{t}\left({F}_{t}^{-1}\left(u\right)\right)$ decreases in u for all $t>0,$ then $\overline{H}\left({T}_{t}\right)\ge \overline{H}\left({T}_{t}^{D}\right)$.

## 4. Jensen–Shannon Divergence of System

**Theorem 7.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The exact value of $\overline{H}\left({T}_{t}\right)$ for various values of k, as demonstrated in Part (b) of Example 1.

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**MDPI and ACS Style**

Kayid, M.; Shrahili, M.
On the Uncertainty Properties of the Conditional Distribution of the Past Life Time. *Entropy* **2023**, *25*, 895.
https://doi.org/10.3390/e25060895

**AMA Style**

Kayid M, Shrahili M.
On the Uncertainty Properties of the Conditional Distribution of the Past Life Time. *Entropy*. 2023; 25(6):895.
https://doi.org/10.3390/e25060895

**Chicago/Turabian Style**

Kayid, Mohamed, and Mansour Shrahili.
2023. "On the Uncertainty Properties of the Conditional Distribution of the Past Life Time" *Entropy* 25, no. 6: 895.
https://doi.org/10.3390/e25060895