# Quantile function after Jacobian transformation

## Main Question or Discussion Point

I am dealing with a random variable which is a transformation of another random variable of the form:

$$Y:=aX^b+c$$

The pdf of the random variable X is known and for the sake of example let it be exponential distribution or any other distribution with known and commonly available quantile function.

If I want to know the median value of Y ,$Y_{M}$, then given that median of X is known and equal to say $X_{M}$ is $Y_{M}$ going to be equal to: $a(X_{M})^b+c$ ?

I suppose I could generate samples from the distribution of Y and use empirical density function to determine approx. quantiles but I'd rather go down the analytical route if possible

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Solved (for the median and I omit c):

$$Pr(Y>y_M)=0.5=Pr(X>x_M)\\ Pr(aX^b>y_M)=0.5\\ Pr\left[X>\left(\frac{y_M}{a}\right)^b\right]=0.5$$

It is true for any percentile of the distribution, just need to replace 0.5 and $y_M$ with appropriate expressions.

Stephen Tashi
I suggest you look at an example like $Y = X^2$ where X has a ramp distribution on the interval $[-1,1]$ given by the probability density $f(x) = \frac{x}{2} + \frac{1}{2}$.
I think the median of $X$ is $\sqrt{2} -1$.