15 Dec '16 19:2810 edits

I have worked out the integral for the mean average, assuming it is defined, of a probability distribution I have discovered, must be;

∫[–∞, ∞] x * |h – u| / ( (h – u)^2 + (x – u)^2 ) dx

h, u, x ∈ ℝ

h, u, x ∈ (–∞, ∞ )

h – u ≠ 0

h and u are the parameters of this distribution while x is the random variable of this distribution.

I have already established that the variable u is both the mode and the median for this distribution (mode = median = u) which, like a normal distribution, has a symmetrical density function curve (of probability_density(x) = |h – u| / ( (h – u)^2 + (x – u)^2 ) ) symmetrical on either side of the mode.

According to WolframAlpha, this integral diverges (implying mean undefined) unless you use a "chuchy principle value" ( https://en.wikipedia.org/wiki/Cauchy_principal_value ) a maths concept I am unfamiliar with and don't understand and seems to often give a mean average contrary to that which I would expect if mean is defined.

But when I use numerical methods using my usual java program for obtaining averages which as always seemed trustworthy in the past, the mean average outputted converges simply on u, which is just what I would expect if this mean is defined.

So

I really want to know because I want to know whether I should say in my book the mean is defined as u or if I should say mean is

∫[–∞, ∞] x * |h – u| / ( (h – u)^2 + (x – u)^2 ) dx

h, u, x ∈ ℝ

h, u, x ∈ (–∞, ∞ )

h – u ≠ 0

h and u are the parameters of this distribution while x is the random variable of this distribution.

I have already established that the variable u is both the mode and the median for this distribution (mode = median = u) which, like a normal distribution, has a symmetrical density function curve (of probability_density(x) = |h – u| / ( (h – u)^2 + (x – u)^2 ) ) symmetrical on either side of the mode.

According to WolframAlpha, this integral diverges (implying mean undefined) unless you use a "chuchy principle value" ( https://en.wikipedia.org/wiki/Cauchy_principal_value ) a maths concept I am unfamiliar with and don't understand and seems to often give a mean average contrary to that which I would expect if mean is defined.

But when I use numerical methods using my usual java program for obtaining averages which as always seemed trustworthy in the past, the mean average outputted converges simply on u, which is just what I would expect if this mean is defined.

So

*is*the mean defined in this case? If so, it is simply u, right?I really want to know because I want to know whether I should say in my book the mean is defined as u or if I should say mean is

*undefined*for this distribution.