Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quantile function after Jacobian transformation

  1. Aug 25, 2012 #1
    I am dealing with a random variable which is a transformation of another random variable of the form:

    [tex] Y:=aX^b+c [/tex]

    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 ,[itex]Y_{M}[/itex], then given that median of X is known and equal to say [itex]X_{M}[/itex] is [itex]Y_{M}[/itex] going to be equal to: [itex]a(X_{M})^b+c[/itex] ?

    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!
     
    Last edited: Aug 25, 2012
  2. jcsd
  3. Aug 26, 2012 #2
    Solved (for the median and I omit c):

    [tex] 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[/tex]

    It is true for any percentile of the distribution, just need to replace 0.5 and [itex]y_M[/itex] with appropriate expressions.
     
  4. Aug 26, 2012 #3

    Stephen Tashi

    User Avatar
    Science Advisor

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

    I think the median of [itex]X [/itex] is [itex] \sqrt{2} -1 [/itex].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Quantile function after Jacobian transformation
Loading...