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:

$$\begin{equation*} \sigma=\sqrt{ \sum_{i=1}^k\limits (x_i - \mu)^2P(x_i) } \end{equation*}$$

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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
$$\begin{eqnarray*} Var(X + Y)& =& Var(X) + Var(Y) + 2Cov(X,Y)\\ Var(X - Y)& =& Var(X) + Var(Y) - 2Cov(X,Y) \end{eqnarray*}$$