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
E(X)=∫∞−∞xf(x)dx=∫∞−∞(c+(x−c))f(x)dx=∫∞−∞cf(x)dx+∫∞−∞(x−c)f(x)dx=c+∫∞−∞(x−c)f(x)dxWe 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.
∫∞−∞(x−c)f(x)dx=∫∞−∞hf(h+c)dh=∫0−∞hf(h+c)dh+∫∞0hf(h+c)dh=−∫∞0(−m)f(c−m)d(−m)+∫∞0hf(h+c)dh=−∫∞0mf(c−m)dm+∫∞0hf(h+c)dh=0Therefore,
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