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I am really at a loss for what this question is even asking. Could someone please explain it to me?

Suppose continuous random variable X has the cumulative distribution function F(x).

Find the CDF G(y) for random variable Y = F(X). Provide all required computational steps.

What computation is there to do here? I'm not really clear.

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This question asks about the distribution of a (random) cumulative distribution function.Here, random cdf is $F(X)$ where $X$ is a random variable having cdf $F$. In this case, $F(X)$ will always have uniform distribution over $(0,1)$. This can be derived very easily by considering the cdf of $F(X)$ and taking inverse cdf approach. One quick intuition is for cdf being a probability, it always lies between $(0,1)$; so is a $U(0,1)$ variate.

For $0<y<1$, $G_Y(y)=\mathbb{P}[Y \le y]=\mathbb{P}[F(X) \le y]=P[X \le F^{-1}(y)]=F(F^{-1}(y))=y$.

Also, note for $y\le0$, $G_Y(y)=0$ and for $y\ge1$, $G_Y(y)=1$.

Thus, $Y=F(X)$ is uniformly distributed over $(0,1)$.

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