Covariance

Covariance is a measure used to determine the extent to which two variables change in tandem. It’s a critical statistic for understanding the relationship between two variables. The covariance ( $Cov (X, Y)$ ) between two variables $X$ and $Y$ is given by:

Cov (X, Y) = E [(X - μ_{X}) (Y - μ_{Y})]

where $μ_{X}$ and $μ_{Y}$ are the means of $X$ and $Y$ , respectively. A positive covariance indicates that the variables tend to move in the same direction, while a negative covariance indicates that they move in opposite directions.

If the (real) random variable pair $(X, Y)$ can take on the values $(x_{i}, y_{i})$ for $i = 1, \dots, n$ , with equal probabilities $p_{i} = \frac{1}{n}$ , then the covariance can be equivalently written in terms of the means $E [X]$ and $E [Y]$ as

Cov (X, Y) = \frac{1}{n} \sum_{i = 1}^{n} (x_{i} - E [X]) (y_{i} - E [Y])

It can also be equivalently expressed, without directly referring to the means,

Cov (X, Y) = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} \frac{1}{2} (x_{i} - x_{j}) (y_{i} - y_{j}) = \frac{1}{n^{2}} \sum_{i} \sum_{j > i} (x_{i} - x_{j}) (y_{i} - y_{j})

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