Can you prove the monotonicity of the function (or find a counter example)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:48:47Z http://mathoverflow.net/feeds/question/95921 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95921/can-you-prove-the-monotonicity-of-the-function-or-find-a-counter-example Can you prove the monotonicity of the function (or find a counter example)? pnifel 2012-05-03T21:33:27Z 2012-05-03T23:27:25Z <p>Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative parameters, where $c &lt; v_h-v_l$. For convenience, define $$a=\frac{v_lk}{v_h\gamma}$$ and $b=k/\gamma$.</p> <p><strong>I am interested in showing that the function $R$ is monotonically increasing in $v_l$ (or finding sufficient conditions on the distribution function that will lead to monotonicity).</strong></p> <p>$$R = v_h(k-\gamma\int_{a}^{b}F(x)dx),$$ where $\gamma \in [0,1]$ is defined implicitly by $$\frac{v_h-v_l}{c}\int_{0}^{a}xf(x)dx=E[x].$$</p> <p>Intermediate results: (1) $\gamma$ is unimodal in $v_l$. Proving monotonicity is straightforward when $\gamma$ is increasing in $v_l$, but is it so when $\gamma$ decreases in $v_l$? (2) $R$ monotonically increases in $v_l$ when $X$ is drawn from a uniform or Pareto distribution with $\alpha > 1$. I also numerically verified that it is the case when $X$ is drawn from a Gamma distribution. </p>