4 ms·
In the mathematical sense this is clearly not true - it’s easy to come up with a smooth single mode distribution that doesn’t look like a bell.
by conformist 2y ago
In the mathematical sense this is clearly not true - it’s easy to come up with a smooth single mode distribution that doesn’t look like a bell.
- thaumasiotes 2y agoIs it? How? You could include a lot of little bells far from the single mode, but that's reading a little too much into the literal meaning of "single mode" - a "bimodal" distribution isn't one where the two most common values are both modes. It's one where there are two distinct local maxima. The tails to the left and right must asymptotically approach zero (or you don't have a smooth distribution, because you have discontinuities somewhere), and if there's just one local maximum, your curve will look like a bell.
- commathingy 2y agoThe exponential distribution (modal value 0) is not bell shaped. If you don't like it's range of non-negative, then take some smooth mollification
- thaumasiotes 2y agoAnd the smooth mollification will look like...?
- pxx 2y agoin the simplest case... just mirror it (some call this a Laplace distribution). if you don't like how it's not differentiable at the mode there are further smoothings (see, e.g., the wikipedia article for this distribution) but this simple construction is continuous.
- canjobear 2y agoIt looks like a spike, not a bell.
- cycomanic 2y agoA spike is not smooth (typically meaning continuous in the variable and its first derivative), which was one of the conditions.
- timy2shoes 2y agoThen take a Cauchy or a t-distribution. Basically anything with a longer tail than exp(x^2). The Gaussian summary will be misleading because of the tails.
- seanhunter 2y agoYes or the chi-square distribution with k=1 or 2 or any other of the gamma distributions[1] with the right parameters will have a shape that is one-sided with the mode at the lower extreme and no "low tail" in the normal sense. [1] https://en.wikipedia.org/wiki/Gamma_distribution https://en.wikipedia.org/wiki/Gamma_distribution or https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/05%3A_Special_Distributions/5.08%3A_The_Gamma_Distribution https://stats.libretexts.org/Bookshelves/Probability_Theory/...
- davidguetta 2y agoThe bell curve IS the smooth single mode with LOWEST ENTROPY.
- nequo 2y agoDo you mean greatest entropy? Not if the support is, for example, the positive reals.