What is the mode of a continuous random variable? Consider a "discrete" random variable $X$. A mode of $X$ is just a maximizer of $P(X = x)$. This is obviously useful, and we can easily see that a mode is a "most likely" value for $X$.
If, instead, we have a "continuous" real-valued random variable $X$ with a PDF $f_{X}$, I think we usually define a mode of $X$ to be a maximizer of $f_{X}$. I have two questions:

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*How can we interpret the mode of a continuous random variable? In other words, why is the mode of a continuous random variable useful to probability theory?

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*Some websites say that it is "the value most likely to lie within the same interval as the outcome", but I can't make sense of that.

*Some people point to the obvious geometric visualization of the mode (as the peak of the PDF) but I don't think that justifies the usefulness of the mode at all.



*Is there a more general definition of mode, removing the assumptions above that $X$ is real-valued and has a PDF?

 A: The intuition for the probability density function $f_X(x)$ of a random variable $X$ is that the chance of $X$ taking values between $a\pm dx/2$ for some very small $dx$ is given by $f_X(a)\cdot dx$. This makes sense in that $\int_{\Omega} f_X(x) dx = 1$ so that the sum of all these "infinitesimal probabilities" is $1$.
So if $f_X(m)$ is the maximal value of $f_X(x)$ then the chance of $X$ taking values between $m\pm dx/2$ is larger that the chance of $X$ taking values in any other interval of length $dx$ (for "small" $dx$). This intuition is what justifies calling $m$ the mode of $X$.
A: Intuitively, the significance of a mode (in the sense of a density maximizer) is that for sufficiently small fixed interval size $\epsilon$, a real-valued random variable $X$ having density $f$ is more likely to realize values in an interval containing the mode than otherwise. Letting $\epsilon\rightarrow 0$ then gives rise to a mode. More formally, by the fundamental theorem of calculus, a mode $m$ satisfies
$$m\in  \arg\max_a \lim_{\epsilon \rightarrow 0}\frac{1}{\epsilon }\int_a^{a+\epsilon}f(x)dx.$$
As for your question on usefulness, finding a mode for a continuous distribution has applications in estimation theory, among other things (e.g. MLE, MAP). Given that a mode captures where data is "most likely" to occur (in the above limit sense), finding a mode gives at least a compelling way to choose an estimator.
