Probability refresher

Some basic definitions and properties needed to understand the material. Here, \(A\) and \(B\) are random variables, and \(\text{E}[\;]\) denotes the expected value operator. \(\alpha\) and \(\beta\) are two real numbers.

CHAIN RULE

\[p(A,B) = p(A) \, p(B|A)\]

INDEPENDENCE of \(A\) and \(B\)

\[p(A,B) = p(A) \, p(B)\]

CONDITIONAL PROBABILITY

\[p(A|B) = \frac{p(A,B)}{p(B)}\]

BAYE'S RULE

\[p(A|B) = \frac{p(B|A) \, p(A)}{p(B)}\]

LINEARITY of \(\text{E}[\;]\)

\[\text{E}[\alpha A + \beta B] = \alpha\text{E}[A] + \beta\text{E}[B]\]

LAW OF TOTAL EXPECTATION

\[\text{E}[A] = \text{E}[\text{E}[A|B]]\]