Expected value and standard deviation

The expected value of a discrete random variabele #X# with #k# possible outcomes equals the mean of the distribution:

\begin{equation*} \mu=\sum_{i=1}^k\limits x_iP(x_i)
\end{equation*}

The standard deviation of this random variable is:
\begin{equation*} \sigma=\sqrt{ \sum_{i=1}^k\limits (x_i - \mu)^2P(x_i) }
\end{equation*}

Expected value and variance of two random variables

Expected value (=mean) of linearly transformed random variable
\begin{equation*}E(aX + b) = aE(X) + b
\end{equation*}
Variance of linearly transformed random variable
\begin{equation*}var(aX + b) = a^2 var(X)
\end{equation*}
Expected value of sum or difference of random variables X and Y
\begin{eqnarray*}
E(X + Y )& =& E(X) + E(Y)\\
E(X - Y )& =& E(X) - E(Y)
\end{eqnarray*}
Variance of the sum or difference of uncorrelated random variables X and Y
\begin{eqnarray*}
Var(X + Y)& =& Var(X) + Var(Y)\\
Var(X - Y)& =& Var(X) + Var(Y)
\end{eqnarray*}
Variance of the sum or difference of correlated random variables X and Y
\begin{eqnarray*}
Var(X + Y)& =& Var(X) + Var(Y) + 2Cov(X,Y)\\
Var(X - Y)& =& Var(X) + Var(Y) - 2Cov(X,Y)
\end{eqnarray*}