Prove that the conditional distribution of a normal random variable is also normal random [duplicate]

How to prove the claim that the conditional distribution of a normal random variable is also normal random? And how to think it intuitively?

• The duplicate addresses the question about intuition. It also shows one way one might prove this claim generally. However, a much simpler way is available: simply inspect the density function. (This is a one-line proof.)
– whuber
Apr 8 '21 at 14:18
• @whuber The link is about bivariate normal distribution. Is there an easier way to think about the simplest case that conditional larger or smaller than a constant? I.e. why is a one-sided truncated normal distribution still "normal"? Apr 8 '21 at 21:35
• I don't follow, because truncated Normal distributions are not Normal, nor are they the same as conditional distributions.
– whuber
Apr 8 '21 at 21:57
• @whuber I find I misunderstand the very basic definition of conditional distribution. Thank you very much for your reply to make me realize that. Apr 9 '21 at 13:08