Covariance Simplified: Learn It Once, Understand It Forever! Video #|109 Data Science and A.I.

Covariance Simplified: Learn It Once, Understand It Forever

Covariance Simplified: Learn It Once, Understand It Forever!

Covariance measures the relationship between two random variables \(X\) and \(Y\). The formula for covariance is:

\[
\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n (x_i – \bar{X})(y_i – \bar{Y})
\]

Expanding the terms:

\[
\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n
\textcolor{red}{x_i y_i} –
\textcolor{green}{x_i \bar{Y}} –
\textcolor{blue}{\bar{X} y_i} +
\textcolor{red}{\bar{X} \bar{Y}}
\]

Simplifying Covariance

We simplify further:

\[
\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n x_i y_i
– \frac{\bar{Y}}{n} \sum_{i=1}^n x_i
– \frac{\bar{X}}{n} \sum_{i=1}^n y_i
+ \frac{\bar{X} \bar{Y}}{n} \sum_{i=1}^n 1
\]

Since \( \sum_{i=1}^n 1 = n \), this simplifies to:

\[
\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n x_i y_i – \bar{X} \bar{Y}
\]

Final Simplification

The covariance formula can also be written as:

\[
\text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n x_i y_i – \left( \frac{1}{n} \sum_{i=1}^n x_i \right) \left( \frac{1}{n} \sum_{i=1}^n y_i \right)
\]

Interpretation of Covariance

Interpretation of the covariance formula:

\[
\text{Cov}(X, Y) =
\textcolor{blue}{\text{(Mean of the product of values of \(X\) and \(Y\))}}

\textcolor{red}{\text{(Product of means of \(X\) and \(Y\))}}
\]

This formula provides computational simplicity and insight into the relationship between \(X\) and \(Y\).

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