Why are Truncated Probability Distributions important in Statistics? Why are Truncated Probability Distributions important in statistics?
Recently, I was reading about "Truncated" Probability Distributions. As the name suggests, a Truncated Probability Distributions is created by taking some Probability Distribution Function and restricting (i.e. "truncating") the range that the random variable can take. However, instead of simply defining the original Probability Distribution Function over a "truncated range" - we end up creating a new Probability Distribution Function over this truncated range (i.e. the Truncated Probability Distribution Function):

My Question: Why is it necessary to create a "Truncated Probability Distribution Function" from the original Probability Distribution Function? Why can't we just restrict the range of the original Probability Distribution Function and perform all our inferences on the original Probability Distribution Function - or is this because doing so would result the in the original Probability Distribution Function not integrating to "1"? Are there any real benefits of using the Truncated Probability Distribution compared to the original Probability Distribution Function? Are there any applications or instances in the real world where it is absolutely necessary to use the Truncated Probability Distribution Function compared to the original Probability Distribution Function?
Thanks!
References:

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*https://en.wikipedia.org/wiki/Truncated_distribution

As you seem to have (almost) guessed, the trucated distribution comes about from imposing the restriction on the support and then multiplying by a scaling constant to make the restricted density integrate/sum to one.  That is all we are doing when we create a truncated version of an initial distribution.
As to when this is useful, it is useful anytime we want to condition on a restricted range for the observable random variable.  This occurs in conditional probability problems when we specify an initial distribution and then condition on the value being in some restricted part of the allowable range.  It also occurs in cases where we use an approximating distribution to approximate another distribution on a smaller support.  Finally, it also occurs in problems with censored data, when we condition on the non-censored part of the data range.
