If the PDF is symmetric about c, show that E(X)=c. This is a homework problem for course STT802-002 Theory of Probabilities and Statistics I in MSU.
Suppose that X has a density $f$ that is symmetric about $c$. That is, $f (c + h) = f (c - h)$ for all real $h$. Show that, if it exists, $E(X) = c$. Hint: Make the change of variable $h = x - c$.
We have already had $c$ in the expression, we just need to prove the second term is 0. Let $h=x-c$, then $dh=dx$, $x=h+c$.
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