...An equivalent relationship holds in terms of random variables.
Relationship between simple linear regression and moments of random variables.
yᵢ = α + βxᵢ + εᵢ, i = 1, 2, ..., n
(where εᵢ represents random noise)
If xᵢ is a random variable that is independently and identically distributed (i.i.d.) for all i=1,2,...,n, and εᵢ (also i.i.d.) has mean 0 and is independent of xᵢ. Then:
β = Cov(xᵢ, yᵢ) / Var(xᵢ)
= Cor(xᵢ, yᵢ) * Sd(yᵢ) / Sd(xᵢ),
α = E(y) - βE(x),
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