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Statistical intervals for truncated continuous distributions

I have written this paper about calculating prediction intervals and confidence intervals for truncated continuous distributions. This is the first mathematical paper I have written(Party) so as well as reviewing my derivation I would appreciate if people could point out if/where I have been using bad practices. For example

1. Using for a non-standard normal distribution.

2. Using f(x) in all 3 sections instead of using a new function for each purpose

3. Labeling equations which I don't make further reference to

4. Saying "cumulative density function" every time which is quite cumbersome, instead of saying "cumulative density function (CDF)" the first time and saying "CDF" every subsequent time.

If I was to expand my work from continuous distributions to discrete distributions the equations would be very similar, would I have to shown a new derivation or could I just say "similarly, for discrete distributions the prediction interval is ..."

Re: Statistical intervals for truncated continuous distributions

I know it's quite an ask to read 7 pages but even those without knowledge in statistics might be able to answer my questions 2,3 and 4.