Consider the following proposition
Proposition C
Let $Z = F(X)$; where $F$ is the continuous cumulative distribution function of the random variable X, then $Z$ has a uniform distribution on $[0, 1]$.
Proof
$$P(Z \leq z) = P(F(X) \leq z) = P(X \leq F^{-1}(z)) = F(F^{-1}(z)) = z$$
This is the uniform cdf.
I can follow the proof above, but my interpretation of its meaning isn't making sense to me. This proof seems to imply that the cdf of any random variable has a uniform distribution. Is this correct? Is there an intuitive explanation for why this is?
For example, consider the graph of the cdf of some normal distributions from wikipedia
How would you map the ideas of this proposition to the graph of the cdf above?
Best Answer
This proposition holds because of the way in which cumulative distribution functions are defined. First note that the proof involves the cdf of $Z = F(X)$, so in the proof we are looking at the cumulative distribution function of a cumulative distribution function. For example, let $z = 0.4$ and consider the plot below
$\hspace{4cm}$
By definition, $P(Z \leq 0.4) = 0.4$, that is, the area under the cdf curve will be $0.4$ by definition.
Also take note of the bounds, if $z < 0$ then $Z = 0$ while if $z > 1$ then $Z = 1$ so $Z$ has a uniform distribution only in [0, 1]