Even central momemt of normal distribution formula
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Since integration is not my strong suit I need some feedback on this, please:
Let YY be N(μ,σ2)N(μ,σ2), the normal distrubutionwith parameters μμ and σ2σ2. I know μμ is the expectation value and σσ is the variance of YY.
I want to calculate the nn-th central moments of YY.
The density function of YY is
f(x)=1σ2π−−√e−12(y−μσ)2f(x)=1σ2πe−12(y−μσ)2
The nn-th central moment of YY is
E[(Y−E(Y))n]E[(Y−E(Y))n]
The nn-th moment of YY is
E(Yn)=ψ(n)(0)E(Yn)=ψ(n)(0)
where ψψ is the Moment-generating function
ψ(t)=E(etX)ψ(t)=E(etX)
So I started calculating:
E[(Y−E(Y))n]=∫R(f(x)−∫Rf(x)dx)ndx=∫R∑k=0n[(nk)(f(x))k(−∫Rf(x)dx)n−k]dx=∑k=0n(nk)(∫R[(f(x))k(−∫Rf(x)dx)n−k]dx)=∑k=0n(nk)(∫R[(f(x))k(−μ)n−k]dx)=∑k=0n(nk)((−μ)n−k∫R(f(x))kdx)=∑k=0n(nk)((−μ)n−kE(Yk))
Let YY be N(μ,σ2)N(μ,σ2), the normal distrubutionwith parameters μμ and σ2σ2. I know μμ is the expectation value and σσ is the variance of YY.
I want to calculate the nn-th central moments of YY.
The density function of YY is
f(x)=1σ2π−−√e−12(y−μσ)2f(x)=1σ2πe−12(y−μσ)2
The nn-th central moment of YY is
E[(Y−E(Y))n]E[(Y−E(Y))n]
The nn-th moment of YY is
E(Yn)=ψ(n)(0)E(Yn)=ψ(n)(0)
where ψψ is the Moment-generating function
ψ(t)=E(etX)ψ(t)=E(etX)
So I started calculating:
E[(Y−E(Y))n]=∫R(f(x)−∫Rf(x)dx)ndx=∫R∑k=0n[(nk)(f(x))k(−∫Rf(x)dx)n−k]dx=∑k=0n(nk)(∫R[(f(x))k(−∫Rf(x)dx)n−k]dx)=∑k=0n(nk)(∫R[(f(x))k(−μ)n−k]dx)=∑k=0n(nk)((−μ)n−k∫R(f(x))kdx)=∑k=0n(nk)((−μ)n−kE(Yk))
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