The nth central moment of X is The skewness of a random vari

The nth central moment of X is The skewness of a random variable measures the lack of symmetry in the PDF and is defined to be . Show that if the PDF of X symmetric about a point x=a then the skewness=0.

Solution

skewness = ₃ /(₂)^3/2

now, ₃ = E [ ( X - E ( X ) )³ ] ..
now, due to symmetry , E(X) = a..so, E( X - a) = E ( a - X)......
So, now ₃ = E [ ( X - a) ³ ]
   = E [(a - X)³ ]
= - E [ ( X - a)³ ]....
So, ₃ = 0..
so, skewness = 0.....