PROOF 172

Prove.

The correlation coefficient lies between −1 and 1

On finite data.

What you may use

A1(u+v)+w = u+(v+w)
associativity of addition
A2u + v = v + u
commutativity of addition
A3u + 0 = u
additive identity
A4u + (−u) = 0
additive inverse
A5c(u + v) = cu + cv
distributive law
A6(uv)w = u(vw)
associativity of multiplication
A7uv = vu
commutativity of multiplication
A8u × 1 = u
multiplicative identity
A9u ≠ 0 ⟹ u·u⁻¹ = 1
multiplicative inverse
A10(u+v)² = u²+2uv+v²
expansion formula (Proof 31)
A11Σ(aᵢ+bᵢ) = Σaᵢ + Σbᵢ
distributivity of sums
A12Σ(k aᵢ) = k Σaᵢ
sum of constant multiples
A130 ≤ t²
squares are non-negative (order)
A14t ≠ 0 ⟹ 0 < t²
squares are positive (order)
A150≤uᵢ, Σuᵢ=0 ⟹ all 0
sum of non-negative terms (order)
A16D>0 ⟹ A(t−α)(t−β), α<β
discriminant and factorization (Proof 101)
A17u ≤ v ⟹ w+u ≤ w+v
order and addition
A18u < v ⟹ w+u < w+v
order and addition (strict)
A190≤u, 0≤v ⟹ 0 ≤ u+v
order and addition (sums)
A200<w ⟹ (u≤v ⟺ wu≤wv)
order and multiplication
A210<u, 0<v ⟹ 0 < uv
order and multiplication 2
A220<u, v<0 ⟹ uv < 0
order and multiplication 3
A23exactly one of u<v, u=v, v<u
trichotomy
A24u<v, v<w ⟹ u<w
transitivity of order
A25u<v ⟹ some w with u<w<v
density (order)
A260 < 1
basic order fact
A270<u ⟹ 0 < u⁻¹
order and reciprocals
A280<u ⟹ 0<√u, (√u)²=u
positive square root (completeness)
A290 ≤ u ⟹ √(u²) = u
square-root convention (Proof 95)
A30(−u)² = u²
sign and squares (Proof 01)
A31uv = 0 ⟹ u=0 or v=0
zero product (Proof 111)
A32u(−v) = −(uv)
sign and products (Proof 81)
Ncomputation on data is determinate
rules of computation
Ea = b ⟹ apply the same operation to both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For sequences a₁, …, aₙ and b₁, …, bₙ, set A = Σaᵢ², B = Σbᵢ², P = Σaᵢbᵢ.

Proof.

Related labs

Correlation lab →