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 F−1(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{x∈R∣F(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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