4.1 Means

The mean measures the ``average'' value.

Call the return on Air.com X, then

$$\begin{eqnarray*}E(X) & = & \textcolor{red}{0.2} \times \textcolor{blue}{-3}\qua... ...tcolor{red}{0.3} \times \textcolor{blue}{3}\quad \\ & = & 0.55 \end{eqnarray*}$$

$$E(X) = \sum_{\textcolor{blue}{x}} \textcolor{red}{P(X = x)}\, \textcolor{blue}{x}.$$

So $1 gains 0.55% on average.

If we had 3 shares (each at a dollar), then the first share gains 0.55, the second 0.55 and the third 0.55, so the total gain is $3 \times 0.55 = 1.65$.

$E(3\, X) = 3 E(X)$

For a general multiplicative weight, which we call w, then

$$\fbox{$E(w \, X) = w E(X).$ }$$