In my assignment, I've came up across this question:
Let $X$ be a discrete random variable which attains the values $x_1 < \dots < x_n$ with probabilities $p_1, \dots , p_n$. With $F$ the cdf of $X$, what is the cdf of $F(X)$?
I know that $F(X)$ is: $$0;(-\infty,x_1)\\ p_1;[x_1,x_2)\\ \vdots\\ p_1+\dots+p_{n-1};[x_{n-1},x_n)\\ 1;[x_n,\infty)$$
Now, from this I am supposed to create another CDF. Is this even possible? I found a solution that goes like this:
$$0;(-\infty,0)\\ \frac{x_1-m}{p-m};[0,p_1)\\ \vdots\\ \frac{x_n-m}{p-m};[p_{n-1},p_n)\\ 1;[p_n,\infty)$$ With $m$ going to minus infinity and $p$ going to plus infinity. I cannot wrap my head around the $p$ and $m$ in the fractions, are they supposed to be "normalizing" the distribution function so it goes to 1? How does it work?