Effect of scaling a probability distribution on median of distribution.

Question

Say a probability distribution has density function $f(x),x\in[a,b]$ and cumulative distribution function as $F(x)$.

Consider the scaled distribution $g(x)=\frac1\theta f(\theta x),x\in[a\theta,b\theta]$ for this is following is the cumulative distribution $$G(x)=\frac{1}{\theta^2}F(\theta x)$$

I want the scaled distribution to have median as $0.99$. That is $$G(0.99)=\frac{1}{\theta^2}F(0.99\times\theta)=0.5$$

Can someone suggest a distribution for which this will always have a parameter fitting to it. $\theta \in [1.01,10000]$

I want a bounded distribution so far I have eliminated

1. beta distribution
2. arcsine distribution
3. Bates distribution
4. Kumarswamy distribution

I have looked at all the distributions mentioned here. https://en.wikipedia.org/wiki/List_of_probability_distributions. Most of the time it is extremely non-trivial to compute the parameters. So I can't really use any given here. Please suggest something if you can think of it.

Answer 1

Your setup is wrong. If you rescale such that the support is on $[a\theta,b\theta]$, then the new density function is $g(x)=\frac1\theta\,f\left(\frac x\theta\right)$ and the new cumulative distribution function is $G(x)=F\left(\frac x\theta\right)$. The new median is simply $m\theta$, where $m$ is the original median.