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:1. 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?2. Is there a more general definition of mode, removing the assumptions above that X is real-valued and has a PDF?

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Consider a &quot;discrete&quot; 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 &quot;most likely&quot; value for   X.If, instead, we have a &quot;continuous&quot; 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:1. 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?2. Is there a more general definition of mode, removing the assumptions above that   X is real-valued and has a PDF?

Tyree Duke · Accepted Answer

Intuitively, the significance of a mode (in the sense of a density maximizer) is that for sufficiently small fixed interval size   ϵ, a real-valued random variable   X having density   f is more likely to realize values in an interval containing the mode than otherwise. Letting   ϵ  →  0 then gives rise to a mode. More formally, by the fundamental theorem of calculus, a mode   m satisfies  m  ∈  arg  ⁡      max    a        lim          ϵ      →      0            1    ϵ        ∫    a          a      +      ϵ        f  (  x  )  d  x  .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 &quot;most likely&quot; to occur (in the above limit sense), finding a mode gives at least a compelling way to choose an estimator.

Camille Flynn · Accepted Answer

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  ±  d  x  /  2 for some very small   d  x is given by       f    X    (  a  )  ⋅  d  x. This makes sense in that       ∫      Ω        f    X    (  x  )  d  x  =  1 so that the sum of all these &quot;infinitesimal probabilities&quot; is   1.So if       f    X    (  m  ) is the maximal value of       f    X    (  x  ) then the chance of   X taking values between   m  ±  d  x  /  2 is larger that the chance of   X taking values in any other interval of length   d  x (for &quot;small&quot;   d  x). This intuition is what justifies calling   m the mode of   X.

Consider a "discrete" random variable X . A mode of X is just a maximizer of P (

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