1. 9 적률 생성 함수와 누적 생성 함수 (Some special Expextations)

1. 9 적률 생성 함수와 누적 생성 함수 (Some special Expextations)

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Def 1.9.1 | (population) mean of r.v. X  : (모집단이 가지는) 평균

 μ := E(X) , = ∫ xf(x) dx

 

Def 1.9.2 | variance of r.v. X  : 분산

 σ² := E[{X-E(X)}²] , = E(X²) - {E(X)}²

 

Remark | E(Xᵏ) : k-th moment, k차 적률

 μ : 1st moment

 σ² : 2nd moment - (1st moment)²

 E[(X-μ)ᵏ) : k-th central moment

 

Def 1.9.3 | Moment Generating Function (적률 생성 함수, mgf)

X : r.v.
Mₓ(t) := E[eᵗˣ], |t|<h, h>0
       = ∫(-∞~∞) eᵗˣf(x) dx

 

Thm 1.9.1 |  Uniqueness of mgf  : mgf가 존재한다면, 유일하다.

 Mₓ(t) : mgf of r.v. X  &  My(t) : mgf of r.v. Y

 ⇒  Fₓ(t) = Fy(t)  ⇔  Mₓ(t) = My(t)

 

Remark 1.9.1 |  mgf may not exist.  : mgf는 존재하지 않을 수도 있다.

 X : r.v. with pdf f(x)=1/x²*I (x>1)

 Mₓ(t) = ∫(1~∞) eᵗˣ1/x² dx = lim(b→∞)∫(1~b) (1+tx+1/2t²x²+· · ·)*1/x² dx  : not integrable

 

Remark 1.9.2 |  Sometimes can find the pdf from the mgf.

 Mₓ(t) = 1/10eᵗ + 2/10e²ᵗ + 3/10e³ᵗ + 4/10e⁴ᵗ

 ⇒ Mₓ(t) = ∑eᵗˣPₓ(x), x=1,2,3,4

 ⇒ Pₓ(1) = 1/10, Px(2) = 2/10, Px(3) = 3/10, Px(4) = 4/10

 ⇒ Pₓ(x) = x/10, x=1,2,3,4

 

Remark 1.9.3 |  Can compute E(Xᵐ), m = 1, 2, · · ·, using the mgf.

Mₓ(t) = E[eᵗˣ] = E[1+tx+1/2t²x²+· · ·]
      = 1 + tE(X) + t²/2*E(X²) + · · ·
      = 1 + μ*t + μ₂²/2*t² + · · ·, where μₖ = E(xᵏ)
      
Mₓ(t)’|t=0 = μ + μ₂t + · · ·|t=0 = μ
Mₓ(t)’’|t=0 = μ₂ + μ₃t + · · ·|t=0 = μ₂
·
·
In general, Mₓ⁽ᵏ⁾(0) = μₖ , k=1, 2, · · ·

i.e. Mₓ(t) = ∑μⱼ/j!*tʲ : power series expansion

 

Remark 1.9.4 |  Characteristic Function (특성 함수, Ch. f)

 φ(t) := E[eⁱᵗˣ], where i : imaginary

 = E[cos(tx) + isin(tx)]

 

· Ch.f always exists

   |φ(t)| = |∫eⁱᵗˣf(x)dx| ≤ ∫|eⁱᵗˣ|f(x)dx = ∫f(x)dx = 1.

· μ = -iφ’(0)

· E(X²) = -φ’’(0)

 

Remark 1.9.5 |  Cumulant generating function (누적 생성 함수, cgf)

 ψ(t) := logMₓ(t) : cgf of r.v. X

 Recall that Mₓ(t) = ∑μⱼ/j!*tʲ : power series , μⱼ : j-th moment

 Let ψ(t) = ∑kⱼ/j!*tʲ : power series , kⱼ : j-th cumulant

 

[Relationship between moments and cumulants]

k₀ = 0, k₁ = μ, k₂ = σ², k₃ = E[(X-μ)³] := μ₃’, · · · 

μⱼ’ = E[(X-μ)ʲ] : j-th central moment

 

Remark 1.9.6 |  Skewness and Kurtosis

· ρ₃ := E[(X-μ)³]/σ³ : skewness(왜도, 치우친 정도)

· ρ₄ := E[(X-μ)⁴]/σ⁴ : kurtosis(첨도, 뾰족한 정도)

 

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