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The 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
by commathingy 2y ago
The 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/...