[Math] Example of a distribution with mode undefined

Question

Is there any distribution with an undefined mode?

If all the elements of the data set have frequency 1 is the mode undefined or all the elements are modes of the distribution?

Answer 1

In my experience, there are at least two ways to define the mode, and the reader has to pay attention to which one an author is using:

(1) For a sample or discrete distribution, the outcome with the highest frequency or probability. For a continuous distribution, the value $x$ at which the PDF $f(x)$ attains its maximum value. Then this is followed by a caveat that the mode need not be unique. (This is the approach in Wikipedia on 'mode'.)

(2) Starts the same, but with the caveat that the mode may not exist. (In the example of your Question, this is probably the most sensible choice.)

This accounts for the differing approaches of two Comments.

Following on from either (1) or (2) one also sees the terminology 'multimodal'. Depending on circumstances described on the spot, there is sometimes an explanation that this does not exactly connect with either definition (1) or (2).

Examples:

(a) The Poisson distribution of random variable $X$ with an integer mean $\lambda > 0$ is said to have a 'double mode' at $x = \lambda$ and $x = \lambda - 1.$ Specifically, with $X \sim Pois(3),$ we have $P(X = 2) = P(X = 3) = 0.22404181.$

(b) Consider a mixture distribution in which a population has 40% members from $Norm(0, 1)$ and 60% from $Norm(3,1)$. One often says that this PDF is 'bimodal' because it has two relative maxima, even though $the$ mode is clearly 3. There are two distinct relative maxima when, as here, the means of the two normal distributions are sufficiently different relative to their standard deviations.

(c) The histogram of a relatively small sample from a umimodal population can appear to have several 'modes', while a larger sample provides a more accurate representation of the population distribution. On the left, $the$ 'modal bin' of the histogram is centered at 115, but there happens to be a 'secondary mode' of the histogram nearer actual the mode of the population PDF (blue curve), which would typically be unknown in practice. The green curves attempt to estimate the PDF from the sampled data--more successfully for the larger sample.

In practice, language using the word 'mode' seems to be adaptable to the circumstances at hand.