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.
Yes 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.
in 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.
Then 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.
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.