Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X

Let X be a random variable with a continuous and strictly increasing c.d.f. F (so that the quantile function F1 is well-defined). Define a new random variable Y by Y=F(X). Show that Y follows a uniform distribution on the interval [0,1].

My initial thought is that Y is distributed on the interval [0,1] because this is the range of F. But how do you show that it is uniform?

Answer

Let FY(y) be the CDF of Y=F(X). Then, for any y[0,1] we have:

FY(y)=Pr.

What distribution has this CDF?

Attribution
Source : Link , Question Author : user162381 , Answer Author : JimmyK4542

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