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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)\\

    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

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    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].
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