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I have asked a related question Probability Mass Function making the Truncated Normal Discrete

I am trying to understand the difference between 2 answers (the difference between 2 PMFs given).

What is the difference (I am mainly interested in descriptive difference and their different properties) between

$$P(X = x\vert \mu,\sigma,a,b,c) = \frac{e^{-\frac{(x-\mu)^2}{2\sigma^2}}}{\sum_{y \in \Omega} e^{-\frac{(y-\mu)^2}{2\sigma^2}}}$$

Where $\Omega = \lbrace a,a+c,a+2c,\dots b-c,c \rbrace$ all the values in the support.

and $$p_X(x) = \frac{\Phi \Big( \frac{x - \mu + c/2}{\sigma} \Big) - \Phi \Big( \frac{x - \mu - c/2}{\sigma} \Big)}{\Phi \Big( \frac{b - \mu + c/2}{\sigma} \Big) - \Phi \Big( \frac{a+c/2 - \mu}{\sigma} \Big)} \quad \quad \quad \text{for } x=a,a+c,a+2c,...b-c,b,$$

One uses the normal density values at the exact points while the other uses the normal probabilities across intervals centred on those points and I can't understand the impications of those differences (entropy, description, fitting).

What are their different properties?

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  • $\begingroup$ @Xi'an Why using the density values at the exact points (simply adjusting it proportionally for the probability to add up to 1) not lead to a discretised Normal? Was Ben's answer the correct one and Tyrel Stokes and Sextus Empiricus wrong; why? $\endgroup$ Nov 13 at 10:58
  • $\begingroup$ @Xi'an 2 points in your comment that I didn't understand. «a discretised version of a continuous distribution is the distribution of a discretisation» it seems really cyclical I don't understand the different meaning (implications) between «a discretised version of a continuous distribution» and «the distribution of a discretisation». Aren't the same thing? The second is «this density could be defined arbitrarily over the integers». What do you mean «defined arbitraly over the integers» and why is that true (you made an affirmation to give justification a reason). $\endgroup$ Nov 13 at 11:24
  • $\begingroup$ Can I please receive some feedback? I do not understand why the question isn't clear. What are the ambiguities in it? $\endgroup$ Nov 18 at 19:47
  • $\begingroup$ This looks like one open-ended question ("what are their different properties") followed by several other questions. A narrower, focused question would fit our criteria better. $\endgroup$
    – whuber
    Nov 20 at 16:51
  • $\begingroup$ @whuber It is not followed up by several other questions (there are only 3 questionmarks and the last 2 are 1 question). I will easily remove the second question (I have a hard time wording it and I can see the confusion created). Open-ended (of a question): allowing the formulation of any answer, rather than a selection from a set of possible answers. And that is the nature the essence of the question, it in not Binary (neither True or False nor A or B) nor Multiple Choice. $\endgroup$ Nov 20 at 17:35