Let X be a scalar random variable. Prove by induction on r that the derivatives of X() and KX(), if they exist, satisfy M (r)

Let X be a scalar random variable. Prove by induction on r that the derivatives of

ΜX(ξ) and KX(ξ), if they exist, satisfy M

(r)

X

(ξ) =

r ∑

j=1

( )M

(r−j)

X

(ξ)K

(j)

X

(ξ) r ≥ 1.

Hence show that

μ

′

r = κr +

r−1

∑

j=1

( )κjμ

′

r−j and, for r ≥ 4, that

μr = κr +

r−2

∑

j=2

( )κjμr−j

(Thiele, 1897, eqn. 22; Morris, 1982).