Chapter 4 Distributions of functions of normally distributed variables Flashcards
(26 cards)
What is the chi-squared distribution and how is it related to the gamma distribution?
The chi-squared (χ²) distribution with ν degrees of freedom is a special case of the gamma distribution: χ²_ν ≡ Gam(ν/2, 1/2).
Example: If X ~ N(0,1), then X² ~ χ²_1 ≡ Gam(1/2, 1/2).
What are the mean and variance of a χ²_ν distribution?
For U ~ χ²_ν: E[U] = ν, Var(U) = 2ν.
Example: For χ²_5, mean=5, variance=10.
How is the χ² distribution used in sampling distributions?
If X₁,…,Xₙ ~ iid N(μ,σ²), then Σ[(Xᵢ-μ)/σ]² ~ χ²ₙ.
Example: For n=10 normal samples, Σ[(Xᵢ-μ)/σ]² follows χ²₁₀.
What is the distribution of the sample variance S² when μ is unknown?
(n-1)S²/σ² ~ χ²ₙ₋₁.
Example: For n=20, 19S²/σ² ~ χ²₁₉ with E[S²]=σ² and Var(S²)=2σ⁴/19.
What is Student’s t-distribution and how is it defined?
T = Z/√(U/ν) where Z~N(0,1), U~χ²_ν, independent.
Example: T = (X̄-μ)/(S/√n) ~ tₙ₋₁ for normal samples.
What are the properties of the t-distribution?
Symmetric about 0, E[T]=0 (ν>1), Var(T)=ν/(ν-2) (ν>2).
Example: t₅ has variance 5/3.
What is the F-distribution and how is it defined?
F = (U/α)/(V/β) where U~χ²_α, V~χ²_β, independent.
Example: For testing σ_X²=σ_Y², S_X²/S_Y² ~ Fₘ₋₁,ₙ₋₁.
What is the relationship between F and chi-squared distributions?
If W ~ F_{α,β}, then 1/W ~ F_{β,α}.
Example: If F ~ F₃,₅, then 1/F ~ F₅,₃.
What is the expected value of an F-distributed random variable?
E[W] = β/(β-2) for β>2.
Example: For F₄,₁₀, E[W]=10/(10-2)=1.25.
How are the sample mean X̄ and sample variance S² related for normal samples?
X̄ and S² are independent for normal samples.
Example: For X₁,…,Xₙ ~ N(μ,σ²), X̄ ⊥ S².
What is the moment generating function of χ²_ν?
MGF of χ²_ν is (1-2s)^(-ν/2).
Example: For χ²₃, MGF=(1-2s)^(-3/2).
What is the PDF of the t-distribution?
f_T(t) = [Γ((ν+1)/2)]/[Γ(ν/2)√(νπ)] * (1+t²/ν)^(-(ν+1)/2).
Example: t₂ has heavier tails than N(0,1).
What is the PDF of the F-distribution?
f_W(w) = (α/β)(αw/β)^(α/2-1)/[B(α/2,β/2)(1+αw/β)^(α+β)/2].
Example: F₁,₁ has PDF f(w)=1/[π√w(1+w)].
How do you construct a t₃ random variable from other distributions?
T = Z/√(U/3) where Z~N(0,1), U~χ²₃, independent.
Example: From X~N(2,9), Y~t₄, U~χ²₃: (X-2)/3 / √(U/3) ~ t₃.
How do you construct an F_{1,4} random variable?
F = (U/1)/(V/4) where U~χ²₁, V~χ²₄.
Example: From Y~t₄: Y² ~ F₁,₄ since t₄ = Z/√(V/4) ⇒ t₄² = Z²/(V/4) ~ F₁,₄.
What is the distribution of nS²_μ/σ² when μ is known?
nS²_μ/σ² ~ χ²ₙ where S²_μ = (1/n)Σ(Xᵢ-μ)².
Example: For n=15, 15S²_μ/σ² ~ χ²₁₅.
Why is the denominator in the t-statistic √(U/ν)?
It scales the χ²_ν variable U to have mean 1 (since E[U/ν]=1).
Example: In t = Z/√(U/ν), denominator adjusts for sample size.
What happens to the t-distribution as ν → ∞?
It approaches N(0,1).
Example: t₁₀₀ is nearly standard normal.
How are the gamma and χ² distributions related?
χ²_ν is Gam(ν/2, 1/2).
Example: Gam(3, 1/2) ≡ χ²₆.
What is the sampling distribution of (X̄-μ)/(S/√n)?
tₙ₋₁ distribution.
Example: For n=10, (X̄-μ)/(S/√10) ~ t₉.
What is the Beta function in the F-distribution PDF?
B(α,β) = Γ(α)Γ(β)/Γ(α+β).
Example: B(1/2,1/2) = π.
How do you verify X² ~ Gam(1/2,1/2) for X~N(0,1)?
Use transformation method or MGF: E[e^{sX²}] = (1-2s)^(-1/2), which matches Gam(1/2,1/2) MGF.
What is the additive property of χ² distributions?
Sum of independent χ²_νᵢ is χ²_Σνᵢ.
Example: χ²₂ + χ²₃ = χ²₅.
How does the F-distribution arise in variance ratio tests?
Under H₀:σ_X²=σ_Y², (S_X²/S_Y²) ~ Fₘ₋₁,ₙ₋₁.
Example: Comparing variances from m=5, n=8 samples gives F₄,₇.
