Relationship between simple linear regression and sample mean, correlation, standard deviation.

Regression equation:
yᵢ = α + βxᵢ + εᵢ, i = 1, 2, ..., n
(where εᵢ represents random noise)

Then the least squares estimator of β is
b = Cor(x, y) * Sd(y) / Sd(x),
and the estimator of α is
a = Mean(y) - b * Mean(x)

(where Mean, Sd, Cor and the sample mean, sample standard deviation, and the sample Pearson correlation coefficient respectively)

...An equivalent relationship holds in terms of random variables.

Relationship between simple linear regression and moments of random variables.

Regression equation:
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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