Wednesday, 16 November 2016

probability - Distribution of continuous distribution function of random variable



Suppose we have X - random variable with distribution function F(x), where F(x) -- continuous distribution function. How to find the distribution function of new random variable Y=F(X)?



Important : There is no assumptions that F(x) is strict monotone. So we can't just find inverse function F1(x).




P.S. I know the answer: U[0;1], but how to prove it? I can prove it only in case, when F(x) - strict monotone.


Answer



Let u(0,1).



If zu:=inf{xRF(x)u}
then F(zu)u since F is right-continuous.



So we have $F(x)

If moreover F is continuous then $P\left(X


Then $P\left(F\left(X\right)

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