Let X be a random variable with the Normal distribution havi

Let X be a random variable with the Normal distribution having mean µ and standard deviation . Show that the nth central moment of X, E{ (X µ )} , µ is equal to 0 when is odd and ( n 1) when n is even (where the double factorial “!!” is defined as k!! ={ k 2} = k(k 2)( k 4) …)

Solution

well as you can see

X- miu is the expected value of X

and it is equal to 0 when is odd and sigma (n-1)!!

n is even