Expected Value of a Discrete Random Variable

Let #X# be a discrete random variable with probability distribution #f(x)# and range #R(X)#.
#\phantom{0}#
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:
\[\begin{array}{rcl}
Var(X + Y)& =& Var(X) + Var(Y) + 2Cov(X,Y)\\
Var(X - Y)& =& Var(X) + Var(Y) - 2Cov(X,Y)
\end{array}\]