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I'm no expert on stats, this discussion is a bit over my head, but it reminds me that not all distributions have a mean. A Cauchy distribution has no mean; as y

I'm no expert on stats, this discussion is a bit over my head, but it reminds me that not all distributions have a mean. A Cauchy distribution has no mean; as you average successive samples, it never settles down to a consistent average. I was surprised when I first encountered Cauchy distributions and up until then had considered that all distributions would have a mean. I suppose it is one of those "heavy-tailed" distributions you're talking about?

SnowHill9902 4y ago
It’s an example but long tailed distributions can have a mean, and short tailed distributions can have no mean. It’s not indicative.
- hanche 4y ago
  > short tailed distributions can have no mean That’s a surprise to me. Now I’m curious: With what definition of “short tailed” can that be true?
oneoff786 4y ago
There’s no need to get mathy tbh. Distributions are not real, and the ones we create only describe parts of what actually happens. This is obvious and always has been. How much water does your faucet leak per day? That might be a good case for a normal distribution. But today I smack it with a hammer and it can’t stop flowing. We now have a ton of water far more than should ever appear on that original normal distribution. Is that because the model is wrong? No. It’s because the model is not designed for these circumstances. We can’t build a model that estimates flow distribution including the possibility that I smack it with a hammer. That doesn’t mean we don’t believe it won’t ever happen or even that we think it is unlikely to happen. It’s just a weird edge case. A lot of people (taleb) imply statisticians are idiots because they insist the distributions are comprehensive. But statisticians don’t insist this. Black swan, fat tails, and all that are just terms for things outside modeled conditions. And yeah. They happen.

SnowHill9902 4y ago

It’s an example but long tailed distributions can have a mean, and short tailed distributions can have no mean. It’s not indicative.

hanche 4y ago
> short tailed distributions can have no mean That’s a surprise to me. Now I’m curious: With what definition of “short tailed” can that be true?

oneoff786 4y ago

There’s no need to get mathy tbh. Distributions are not real, and the ones we create only describe parts of what actually happens. This is obvious and always has been. How much water does your faucet leak per day? That might be a good case for a normal distribution. But today I smack it with a hammer and it can’t stop flowing. We now have a ton of water far more than should ever appear on that original normal distribution. Is that because the model is wrong? No. It’s because the model is not designed for these circumstances. We can’t build a model that estimates flow distribution including the possibility that I smack it with a hammer. That doesn’t mean we don’t believe it won’t ever happen or even that we think it is unlikely to happen. It’s just a weird edge case. A lot of people (taleb) imply statisticians are idiots because they insist the distributions are comprehensive. But statisticians don’t insist this. Black swan, fat tails, and all that are just terms for things outside modeled conditions. And yeah. They happen.