gnome
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The "first central moment" of a real-valued function
\mu_1 \equiv \int_{-\infty}^\infty (x - \mu) f(x)\,dx = 0
where
\mu \equiv \int_{-\infty}^\infty x\, f(x)\,dx
so we have
\int_{-\infty}^\infty (x - \left ( \int_{-\infty}^\infty x\, f(x)\,dx \right ) ) f(x)\,dx = 0
Intuitively, it seems to make sense, but how do we manipulate those integrals to prove this equality?
\mu_1 \equiv \int_{-\infty}^\infty (x - \mu) f(x)\,dx = 0
where
\mu \equiv \int_{-\infty}^\infty x\, f(x)\,dx
so we have
\int_{-\infty}^\infty (x - \left ( \int_{-\infty}^\infty x\, f(x)\,dx \right ) ) f(x)\,dx = 0
Intuitively, it seems to make sense, but how do we manipulate those integrals to prove this equality?