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When alpha is 2 or less than 2 the variance of the inverse gamma doesn't exist. That is why when the variance is defined for the inverse gamma it always says "for α > 2". It is also the case that when alpha is 1 or less the mean of the inverse gamma doesn't exist. In order to really undertand what it means to say the variance doesn't exist (or the mean doesn't exist) you need to understand the mathematical definition of the variance (and of the mean). I don't know how to add the necessary symbols to clearly explain this. However, just briefly, mathematically both the mean and variance of the gamma density are definite integrals over the support of the density, which is 0 to infinity. In general, sometimes a definity integral over an infinite range (negative and/or positive) exists and sometimes it doesn't. In the case of the definite integral for the variance on the inverse gamma, when alpha less than or equal to 2, this integral doesn't exist.

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Q: What is the variance of a inverse gamma distribution with alpha equals 2?
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