Chapter 7 Linear transformations

We already know that $𝖤\left[aX\right]=a𝖤\left[X\right]$ and $\mathrm{𝖵𝖺𝗋}\left[aX\right]=a^2\mathrm{𝖵𝖺𝗋}\left[X\right]$. From the previous chapter, linearity of expectation gives $𝖤\left[aX+bY\right]=a𝖤\left[X\right]+b𝖤\left[Y\right]$, but what about $\mathrm{𝖵𝖺𝗋}\left[aX+bY\right]$? In this Chapter, we examine expectations and variances of linear combinations of random variable, starting with two random variables and then extending to $n$ random variables:

We start by introducing the covariance between any two random variables $X$ and $Y$; this is of interest in its own right as a measure of the dependence between $X$ and $Y$, and it it leads to the concept of correlation. In general, covariance is also the key additional ingredient required to evaluate $\mathrm{𝖵𝖺𝗋}\left[aX+bY\right]$ .