Let X be a continuous random variable having cumulative distribution function F. Define the ran- dom variable Y by Y

The likelihood that a real-valued random variable, let's say "X," would have a value less than or equal to the value x is represented by the cumulative distribution function (CDF) of that variable in probability and statistics.
A random variable is one that specifies the potential values of an unexpected phenomenon's outcome. It has definitions for both random and discrete variables. The distribution of the multivariate random variables is also specified using it.
The tail distribution, sometimes referred to as the complementary cumulative distribution function, occurs when the random variable is above a certain level (CCDF). You will learn about the cumulative distribution function in this article, along with its characteristics, calculations, uses, and examples.

Let [tex]X[/tex] be a continuous random variable with CDF [tex]F[/tex].

Now, define [tex]Y=F(X)[/tex]. Since we have that [tex]X[/tex] ∈ [tex]R[/tex], we will also have [tex]F(X)[/tex]∈[tex][0,1][/tex]. So, take any [tex]y[/tex]∈[0,1]. We have that,

[tex]P[/tex][tex]([/tex][tex]Y[/tex]≥[tex]y[/tex][tex])[/tex][tex]=[/tex][tex]P(F(X)[/tex]≤[tex]y[/tex][tex]=[/tex][tex]P(X\leq F^{-1} (y))[/tex][tex]=F(F^{-1}(y))[/tex][tex]=y[/tex]

So, we see that the CDF of [tex]Y[/tex] is the function [tex]F_{Y} (y)[/tex][tex]=y[/tex] on the interval [tex][0,1][/tex].

Hence, we have: [tex]Y[/tex]∼Unif[tex](0,1)[/tex].

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