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For $a_1,a_2$, $b_1,b_2$ $\in\mathbb{R}^+$, if $a_1 , then for any perturbation $\epsilon\in \mathbb{R}^+$, $$r_1=\frac{a_1+\epsilon}{b_1+\epsilon}>\frac{a_1}{b_1} $$ and if $a_2>b_2$, $$r_2=\frac{a_2+\epsilon}{b_2+\epsilon}<\frac{a_2}{b_2} $$

If $\epsilon$ is generated from a continuous function as $\epsilon={f_\epsilon(t)}$. What would be a way to characterize $r_1-r_2$ in terms of $f_\epsilon(t)?$

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    I don't know what $f_{\epsilon}(t=x)$ means.2012-03-06
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    Also, I don't what $x$ and $y$ have to do with $a_1,a_2,b_1,b_2$.2012-03-06
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    This is a non-stochastic version of my question on http://stats.stackexchange.com/questions/24212/perturbations-of-rational-numbers-with-random-variables . I have edited the question-now. Basically, the $\epsilon$ is a r.v in that version(on stats stack exchange) of the question with $f_\epsilon$ being a one-parameter pdf from an exponential family. Over here, it is a continuous function with no probabilistic assumptions.2012-03-06
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    Sorry, still don't understand. So $\epsilon$ is a continuous function of $t$ - but what does $t$ have to do with $a_1,a_2,b_1,b_2$?2012-03-06
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    A rational number is shrinking while the other is getting larger with this form of a perturbation. $t$ is not related to $a_1,a_2$, $b_1,b_2$. I would like to see how the difference between the rationals varies w.r.t the function in one-varible, $t$. $f_\epsilon$ can be linear/monotonic or say, a higher order polynomial. I do agree that the class of functions, being referred to-with $f_\epsilon$ has not been specified. As $\epsilon$ asymptotically gets larger the ratios tend to converge to 1. How does the rate of change of $r_1$ and $r_2$ relate to the rate of change of $f_\epsilon$ ?2012-03-07

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