Given $X$ be a random variable and let $f$ be an increasing function and $Y$ be another random variable with $Y=f(X)$. What is the relation between medians of $X$ and $Y$?
I tried the above by finding the CDF of distribution defined by $Y$ as below:
$F_{Y}(y)=P(Y \leq y)$ where $y$ is the median
$F_{Y}(y)=P(f(X) \leq y)$
$F_{Y}(y)=P(X \leq f^{-1}(y))$ as function $f$ is strictly increasing
$F_{Y}(y)=F_{X}(f^{-1}(y))$
Thus, if $y$ is the median for distribution defined by $Y$, then $f^{-1}(y)$ is the median for distribution defined by $X$ or it can be represented as $y=f(x)$ where $y$ and $x$ are medians of the distribution defined by $Y$ and $X$.
Can you please let me know if my approach to the above formulation is correct?
For me your reasoning is correct. In general, if we call $q_{\alpha}^Z$ the $\alpha$ quantile of a generic r.v. $Z$, this respects:
$\alpha=F_Z(q_{\alpha}^Z)$
Now let's consider $Y=f(X)$. Using your last equation for $y=q_{\alpha}^Y$:
$\alpha=F_Y(q_{\alpha}^Y)=F_X(f^{-1}(q_{\alpha}^Y))=F_X(q_{\alpha}^X)$
therefore:
$q_{\alpha}^X=f^{-1}(q_{\alpha}^Y)$
and:
$q_{\alpha}^Y=f(q_{\alpha}^X)$
For $\alpha=1/2$ you get your mapping between medians.