Expected Value of a Discrete Random Variable

Let $X$ be a discrete random variable with probability distribution $f(x)$ and range $R(X)$.
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Then the expected value of $X$ is calculated as follows:
$$\mu_{\small{X}} = \mathbb{E}[X]=\sum_{\text{all }x\text{ in }R(X)}x\cdot f(x)$$

where $f(x)=\mathbb{P}(X=x)$.

Variance and Standard Deviation of a Discrete Random Variable

Let $X$ be a discrete random variable with expected value $\mathbb{E}[X]$.
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Then the variance of $X$ is calculated as follows:

$$\sigma^2_{\small{X}} = Var[X] =\mathbb{E}[X^2] - (E[X])^2$$
To calculate the standard deviation, simply take the positive square root of the variance:

$$\sigma_{\small{X}}= SD[X] = \sqrt{Var[X]}$$

Linear Transformations

The expected value of a linearly transformed random variable is caluclated as follows:
$$\mathbb{E}[aX + b] = a \cdot \mathbb{E}[X] + b$$
The variance of linearly transformed random variable is calculated as follows:
$$Var[aX + b] = a^2 \cdot Var[X]$$
The standard deviation of a linearly transformed random variable is calculated as follows:
$$SD[aX + b] = |a| \cdot SD[X]$$

where $|a|$ is the absolute value of $a$.

Sums and Differences of Random Variables

The expected value of the sum or difference of two random variables $X$ and $Y$ is calculated as follows:
$$\begin{array}{rcl} E(X + Y )& =& E(X) + E(Y)\\ E(X - Y )& =& E(X) - E(Y) \end{array}$$
The variance of the sum or difference of two independent random variables $X$ and $Y$ is calculated as follows:
$$\begin{array}{rcl} Var(X + Y)& =& Var(X) + Var(Y)\\ Var(X - Y)& =& Var(X) + Var(Y) \end{array}$$
The variance of the sum or difference of two dependent random variables $X$ and $Y$ is calculated as follows:
$$\begin{array}{rcl} Var(X + Y)& =& Var(X) + Var(Y) + 2Cov(X,Y)\\ Var(X - Y)& =& Var(X) + Var(Y) - 2Cov(X,Y) \end{array}$$