1. The problem is to understand and use the formula for covariance. 2. Covariance measures how two random variables change together. The formula for covariance between two variables $X$ and $Y$ is: $$\text{Cov}(X,Y) = E[(X - E[X])(Y - E[Y])]$$ where $E$ denotes the expected value (mean). 3. Another common formula for covariance using sums for sample data is: $$\text{Cov}(X,Y) = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})$$ where $n$ is the number of data points, $X_i$ and $Y_i$ are individual data points, and $\bar{X}$ and $\bar{Y}$ are sample means. 4. Important rules: - Covariance can be positive, negative, or zero. - Positive covariance means variables tend to increase together. - Negative covariance means one variable tends to increase when the other decreases. - Zero covariance means no linear relationship. 5. To calculate covariance: - Find the mean of $X$ and $Y$. - Subtract the means from each data point. - Multiply the differences for corresponding $X$ and $Y$ values. - Sum these products. - Divide by $n-1$ for sample covariance. This formula helps understand the relationship between two variables.