Is the following true: A distribution does not have a well defined mean iff its mean comes out as $pminfty?$
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I often heard that some probability distributions do not have a mean. However, the mean is just the expectation value of the distribution's random variable: $mu = leftlangle xf_{theta}(x) rightrangle$ . Here, $theta$ is the distribution's parameter, and $x$ is the random variable. For any probability density function this is an integral which I can compute. So saying that an integral like the above expression is not well defined is basically equivalent to saying that it diverges. Is this correct, or are there other ways in which a distribution can not have a well-defined mean?
probability-theory probability-distributions
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