28 Jan '16 21:14>7 edits
Originally posted by twhiteheadAs I understand it, the mode of a probability distribution is, by the very definition of mode, the value of the random variable of the distribution that has the highest probability. But to be more accurate in the case of a continuous distribution, the mode is the value of the random variable that has the greatest probability density (which itself is not a true probability because you can only get a true probability over some integral of the probability density function ).
I really don't see why you want to name it different or treat it different. Mode is somewhat loosely defined anyway, I see no reason why you shouldn't just extend it to include limits that aren't members of the set.
See this apparently confirmed at:
https://en.wikipedia.org/wiki/Mode_%28statistics%29
"...
The mode of a continuous probability distribution is the value x at which its probability density function has its maximum value, ..."
But if that said mode has a value that the random variable has exactly zero probability density then I don't see how it would comply with that definition of mode. That's why I think there may be a need to, not to say it isn't a mode because I changed my mind about that since my OP, but rather say it is a different 'kind' of mode that doesn't exactly fit with the usual formal meaning of a mode.
I am afraid I am always inclined to be rather pedantic like that.