B3 - multiple random variables and join probabilities

the joint probability mass function between discrete random variables $X$ and $Y$ is defined by:

p_{X, Y} (x, y) = P {X = x & Y = y}

this can also be written as: $P {X = x, Y = y}$ or $P {X = x \cap Y = y}$ .
by definition:

\sum_{x} \sum_{y} p_{X, Y} (x, y) = 1

by the law of total probability:

\begin{matrix} p_{X} (x) = \sum_{y} p_{X, Y} (x, y) \\ p_{Y} (y) = \sum_{x} p_{X, Y} (x, y) \end{matrix}

written as such, $p_{X} (x)$ is often referred to as the marginal probability mass function

independence

it is said discrete random variables $X$ and $Y$ are independent, ie $X ⊥ Y$ , if:

P {X = x & Y = y} = P {X = x} \cdot P {Y = y}, \forall x, y

or, equivalently,

p_{X, Y} (x, y) = p_{X} (x) \cdot p_{Y} (y)

law of total probability

$P (E) = \sum_{y} P (E \cap Y = y) = \sum_{y} P (E ∣ Y = y) P (Y = y)$ $P (X = k) = \sum_{y} P (X = k \cap Y = y) = \sum_{y} P (X = k ∣ Y = y) P (Y = y)$