General Addition Rule for Variances (Perfect Positive Correlation) Confused on the general addition rule for variances.
Why is it that when two random variables, X and Y, have a perfect positive correlation (p=1) their standard deviations add. But when they are uncorrelated (p=0) their variances add? 
 A: The equation $\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2 \text{Cov}(X,Y)$ covers all cases.  Note that $ \text{Cov}(X,Y) = \rho\; \sigma_X\; \sigma_Y$.
Where it comes from is the linearity of expected value:
$$\eqalign{E[X+Y] &= E[X] + E[Y]\cr E[(X+Y)^2] &= E[X^2 + 2 X Y + Y^2] = E[X^2] + 2 E[XY] + E[Y^2]\cr}$$
and the definitions of variance and covariance in terms of expected values.
A: Further illumination on Robert Israel's answer:
Starting with the formula
$$
\begin{align}
\mathrm{Var}(X)
&=\mathrm{E}\left((X-\mathrm{E}(X))^2\right)\\
&=\mathrm{E}(X^2)-\mathrm{E}(X)^2\\
\end{align}
$$
we get
$$
\begin{align}
\mathrm{Var}(X+Y)
&=\mathrm{E}\left(X^2+2XY+Y^2\right)-\left(\mathrm{E}(X)^2+2\mathrm{E}(X)\mathrm{E}(Y)+\mathrm{E}(y)^2\right)\\
&=\left(\mathrm{E}(X^2)-\mathrm{E}(X)^2\right)+\left(\mathrm{E}(Y^2)-\mathrm{E}(Y)^2\right)+2\Big(\mathrm{E}(XY)-\mathrm{E}(X)\mathrm{E}(Y)\Big)\\[5pt]
&=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\mathrm{Cov}(X,Y)
\end{align}
$$
Furthermore, using Hölder,
$$
\begin{align}
\left|\mathrm{Cov}(X,Y)\right|^2
&=\left|\mathrm{E}\Big((X-\mathrm{E}(X))(Y-\mathrm{E}(Y))\Big)\right|^2\\
&\le\mathrm{E}\left((X-\mathrm{E}(X))^2\right)\mathrm{E}\left((Y-\mathrm{E}(Y))^2\right)\\
&=\mathrm{Var}(X)\mathrm{Var}(Y)
\end{align}
$$
For perfect positive correlation, we have $\mathrm{Cov}(X,Y)=\mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}$; therefore,
$$
\begin{align}
\mathrm{Var}(X+Y)
&=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}\\
\mathrm{Std}(X+Y)^2
&=\mathrm{Std}(X)^2+\mathrm{Std}(Y)^2+2\mathrm{Std}(X)\mathrm{Std}(Y)\\
\mathrm{Std}(X+Y)
&=\mathrm{Std}(X)+\mathrm{Std}(Y)
\end{align}
$$
For independence, we have $\mathrm{Cov}(X,Y)=0$; therefore,
$$
\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)
$$
