Let X be a scalar random variable. Prove by induction on r that the derivatives of X()
Question:
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).
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Related Book For
Tensor Methods In Statistics Monographs On Statistics And Applied Probability
ISBN: 9781315898018
1st Edition
Authors: Peter McCullagh
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