Quantile function after Jacobian transformation

  • #1

Main Question or Discussion Point

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:

Answers and Replies

  • #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.
 
  • #3
Stephen Tashi
Science Advisor
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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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