# Find the Expectation of a Symmetric Probabiliry Density Function

2017-10-18

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.

# Problems

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 $h = x - c$.

# Solution 1

We have already had $c$ in the expression, we just need to prove the second term is 0. Let $h=x-c$, then $dh=dx$, $x=h+c$.

Therefore,