- #1
Bipolarity
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I recall reading somewhere that the mean value of a continuous variable is situated at a point that acts as a fulcrum about which all other values are considered "weights".
In other words, if we define the mean as
[tex] μ = \int^{∞}_{-∞} x ρ(x) dx [/tex] (where rho is the probability density)
then can we prove that
[tex] \int^{∞}_{μ} |x-μ| ρ(x) dx = \int^{μ}_{-∞} |x-μ| ρ(x) dx [/tex]
I am not sure my question is very clear considering I don't understand this too well, but perhaps someone understands what I mean?
I'm also not sure the equation is even correct, but my memory tells me I did this a while ago and my gut tells me my memory is not wrong. :D
EDIT: Made a big error with the LaTeX. Just fixed it.
BiP
In other words, if we define the mean as
[tex] μ = \int^{∞}_{-∞} x ρ(x) dx [/tex] (where rho is the probability density)
then can we prove that
[tex] \int^{∞}_{μ} |x-μ| ρ(x) dx = \int^{μ}_{-∞} |x-μ| ρ(x) dx [/tex]
I am not sure my question is very clear considering I don't understand this too well, but perhaps someone understands what I mean?
I'm also not sure the equation is even correct, but my memory tells me I did this a while ago and my gut tells me my memory is not wrong. :D
EDIT: Made a big error with the LaTeX. Just fixed it.
BiP
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