A #statistics question:
Let's say we have a symmetric distribution around /(x=\alpha/). In this case is the expected value /(/mathbb{E}(X)=\alpha/)?
If the symmetric distribution is /(f(x)/), then what would be the median? Is it /(f(/alpha)/) or is it just /(/alpha/).
@BayesicTony good point, would it be defined on a finite interval?
@barefootstache Not sure what you mean by this, but Wikipedia covers the basics:
https://en.wikipedia.org/wiki/Expected_value
@BayesicTony as you see by the Cauchy distribution's mean is from \(]-\infty, \infty[\) thus the value from the integral would be \(\infty-\infty\), which is undefined. Though if one replaced those borders with actual numbers, the integral would be defined.
@barefootstache But then it is not a Cauchy distribution any more, and not a proper density either.
What I was trying to say is, Cauchy distribution is the counter-example that shows that the answer to your question about the expected value is "in general, no."
@barefootstache Cauchy is symmetric, but the mean is undefined.