C Statistical Inference
This section provides a brief introduction into concepts of probability theory and statistical inference that are essential for understanding the technical parts of the main text.
C.1 Distributions
C.1.1 Single Random Variable
The expected value of a discrete random variable \(X\) is the weighted avarage of the possible values \(x\), with their probabilites as weights
\[\begin{equation} E(X) = \sum_{x} p(x) x \end{equation}\]
The expected value of a sum of two random variables is the sum of their expected values
\[\begin{equation} E(X + Y) = E(X) + E(Y) \end{equation}\]
and the expected value scales linearly with scaling factor \(a\)
\[\begin{equation} E(a X) = a E(X) \end{equation}\]
Note, however, that this property does not hold for the product of two random variables
\[\begin{equation} E(X \cdot Y) = E(X) \cdot E(Y) \end{equation}\]
only if \(X\) and \(Y\) are independent.
C.1.2 Multiple Random Variables
- Covariance
- Correlation
- Conditional Probability
- Regression Parameter