Joint Distribution

Parameters

Measure of the direction of linear relationship between 2 variables. Depends on the scale and unit of the variables.

Discrete version:

\mathrm{Cov}(X, Y) = \sum_{x} \sum_{y} (x - \mu_X)(y - \mu_Y) \cdot P(X = x, Y = y)

Continuous version:

\mathrm{Cov}(X, Y) = \int_{-\infty}^\infty \int_{-\infty}^\infty (x - \mu_X)(y - \mu_Y) f(x, y) \, dx \, dy

Sign of covariance	Description
Positive	Variables increase together
Negative	One increases as other decreases
Zero	No linear relationship

Measure of strength and direction of linear relationship between 2 variables. Always in the range $[-1,1]$ .

\rho = \frac{\mathrm{Cov}(X, Y)}{\sigma_X \sigma_Y}

\mathrm{cov}(X, Y) = \frac{1}{n - 1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})

r = \frac{\mathrm{cov}(X, Y)}{s_X s_Y} = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2} \cdot \sqrt{\sum_{i=1}^n (y_i - \bar{y})^2}}

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September 29, 2026 1 min read

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