# Fuzzy Random Walkers with Second Order Bounds: An Asymmetric Analysis
## 3 Edge-Fuzzy Graphs
### Reciprocal Random Variables  
Because of the Definitions 2 and 3 for \(\delta(e_{k})\) and \(\Delta\big{(}p_{j}\big{)}\), respectively, the properties of an inverse random variable gain more interest. The following definition is straightforward. Definition 4: The inverse distribution of a mass distribution function of a random variable \(X\) is defined as the mass distribution function of \(\frac{1}{X}\)[53]. Property 1: In the continuous case, the distributions of \(X\) and \(\frac{1}{X}\) are linked as  
\[f_{\frac{1}{X}}(y)\;=\;\frac{1}{y^{2}}\,f_{X}\bigg{(}\frac{1}{y}\bigg{)},\quad y \;\
eq\;0. \tag{10}\]  
Proof: The cumulative distribution of \(\frac{1}{X}\) is defined as  
\[F_{\frac{1}{X}}(y)\stackrel{{\triangle}}{{=}}\text{prob}\left\{ \frac{1}{X}\;\leq\;y\right\}\;=\;\text{prob}\left\{X\;\geq\;\frac{1}{y}\right\} \;=\;1-\text{prob}\left\{X<\frac{1}{y}\right\}\;=\;1-F_{X}\bigg{(}\frac{1}{y} \bigg{)}. \tag{11}\]  
By differentiating the last relationship, the stated result follows. For instance, if \(X\) is the continuous uniform random variable in \([\alpha_{1},\alpha_{2}]\), where \(\alpha_{1},\alpha_{2}\
eq 0\), then the distribution of \(\frac{1}{X}\) is  
\[f_{\frac{1}{X}}(y)\;=\;\frac{1}{y^{2}(\alpha_{2}-\alpha_{1})},\quad y\in\left[ \frac{1}{\alpha_{2}},\frac{1}{\alpha_{1}}\right]. \tag{12}\]  
Despite its simplicity, in certain scenaria, relationship (10) cannot be used. For instance, only the first moments of \(X\) may be known or \(y=0\) might be a legitimate value, in which case there is a singularity in the inversion of \(f_{X}(x)\). Instead, bounds are sought for the first moments of \(\frac{1}{X}\), which makes more sense from a programming viewpoint in the case of large graphs. Jensen inequality provides a straightforward way to bound the expected value \(\operatorname{E}\left[\frac{1}{X}\right]\) by using the expected value \(\operatorname{E}\left[X\right]\) of the non-zero random variable \(X\). Theorem 1: (Jensen inequality) For any random variable and any convex function \(g(x)\) provided that both the domains of \(X\) and \(\operatorname{E}\left[X\right]\) are subsets of the domain of \(g(\cdot)\)  
\[g(\operatorname{E}\left[X\right])\;\leq\;\operatorname{E}\left[g(X)\right]. \tag{13}\]  
**Corollary 1**. : _The mean value of the strictly positive random variable \(\frac{1}{X}\) has a lower bound of_  
\[\frac{1}{\operatorname{E}\left[X\right]}\;\leq\;\operatorname{E}\left[\frac{1}{X }\right],\quad X\;\
eq\;0. \tag{14}\]  
**Property 2**. : _Function \(g(x)\;=\;\frac{1}{x}\) is convex when \(x>0\)._  
Proof. : Notice that the second derivative \(g^{(2)}(x)=\frac{2}{x^{3}}\) is positive when \(x\) is positive. An alternative way to prove this claim is to apply the standard convexity definition. For every \(\alpha_{0}\in[0,1]\), \(x_{1}>0\), and \(x_{2}>0\)  
\[g(\alpha_{0}x_{1}+(1-\alpha_{0})x_{2})\;\leq\;\alpha_{0}g(x_{1}) +(1-\alpha_{0})g(x_{2}) \Rightarrow\] \[\frac{1}{\alpha_{0}x_{1}+(1-\alpha_{0})x_{2}}\;\leq\;\frac{ \alpha_{0}}{x_{1}}+\frac{1-\alpha_{0}}{x_{2}}\] \[\alpha_{0}x_{2}(\alpha_{0}x_{1}+(1-\alpha_{0})x_{2})+(1-\alpha_ {0})x_{1}(\alpha_{0}x_{1}+(1-\alpha_{0})x_{2})-x_{1}x_{2}\;\geq\;0 \Leftrightarrow\] \[\alpha_{0}(1-\alpha_{0})(x_{1}-x_{2})^{2}\;\geq\;0. \tag{15}\]  
In order to derive realistic upper bounds for the path lengths of a given fuzzy graph, certain probabilistic inequalities can be employed. The first is Markov inequality, which establishes a first order bound for the probability of \(X\) taking very large values by stating that  
**Theorem 2**. : _(Markov inequality) The probability of a strictly positive random variable \(X\) exceeding \(\gamma_{0}\) is bounded by_  
\[\operatorname{prob}\left\{X\geq\gamma_{0}\right\}\;\leq\;\frac{\operatorname{ E}\left[X\right]}{\gamma_{0}},\quad\gamma_{0}>0,X>0. \tag{16}\]  
Second order bounds can be derived by the Chebyshev inequality. The latter provides tighter bounds while lifting the positivity assumption. **Theorem 3**.