Quantile function after Jacobian transformation

[WantToBeSmart]

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

Thanks in advance!

 

[WantToBeSmart]

Solved (for the median and I omit c):

Pr(Y&gt;y_M)=0.5=Pr(X&gt;x_M)\\<br /> Pr(aX^b&gt;y_M)=0.5\\<br /> Pr\left[X&gt;\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.