Sums of independent random variables and limit theorems

The distribution of X+Y with X and Y are real valued, independent discrete random variables P(X+Y = Z) = Σₓ P(X=x, Y=Z-X) P(X+Y = Z) = Σₓ P(X=x)* P(Y=Z-X) pₓ+y(z) = Σₓ pₓ(x) py(z-x) fₓ+y(z) = -∞∫∞ fₓ(x)*fy(z-x) dx

X~Γ(α, λ) α, λ>0 fₓ(x): • (λᶛ /Γ(α))(e^(λx))x^(α-1) x>0 • 0 otherwise Where Γ(α) is the so-called Euler Γ-function Γ(α)= ₀∫∞ (e^(λx))*x^(α-1) dx In particular if X₁, X₂, …, Xₙ ~ Exp(λ) is independent then Xᵢ ~Γ(1, λ) , X₁, X₂, …, Xₙ ~Γ(n, λ)

Consider X₁, X₂, …, Xₙ random variables, real valued independent and with the same distribution (iid) E(Xᵢ) = E[(X₁, X₂, …, Xₙ)/n] = µ Var(Xᵢ) = ͳ² * (1/n) So the distribution of Xₙ gets more and more concentrated around of µ as n –> + ∞

For every ε>0 lim P(|Xₙ - µ|)⩾ε) = 0 Roughly speaking Xₙ = µ for n large

E(X₁ + X₂ + … + Xₙ) = nµ Var(X₁ + X₂ + … + Xₙ) = nͳ² Define Zₙ = (X₁ + X₂ + … + Xₙ) / (ͳ * √n) E(Zₙ) = 0 Var(Zₙ) = 1 Remark ▪ Suppose X~N(µ, ͳ²) Then Zₙ~N(0, 1) ▪ Zₙ = (Xₙ - µ) / (ͳ / √n) So Xₙ = µ + Zₙ * (ͳ /√n) Where Zₙ * (ͳ /√n) is the error

As n –> ∞ the distribution of Zₙ approaches the distribution N(0, 1) lim P(Zₙ ⩽ x) = Ф(x) ∀x Ф(x) –> Distrib. function of a N(0, 1)

P(a⩽ (X₁ + X₂ + … + Xₙ) ⩽ b = = P(a - nµ / (ͳ * √n) ⩽ (X₁ + X₂ + … + Xₙ- nµ ) / (ͳ * √n) ⩽ b - nµ/ (ͳ * √n)) = = P(a-nµ / (ͳ * √n) ⩽ Zₙ ⩽ b - nµ/ (ͳ * √n)) = = Ф(b - nµ/ (ͳ * √n)) - Ф(a - nµ / (ͳ * √n)) If low n P = Ф(b + 1/2 - nµ/ (ͳ * √n)) - Ф(a - 1/2 - nµ / (ͳ * √n)) –> correction of continuity Example X~Be(p) need correction if np >5 n(1-p) >5