Independence and Conditional Independence

Independence

• Random variables $X$ and $Y$ are not independent, i.e., $X \not\perp\!\!\!\perp Y$, if there exists two values of $Y$, say $y_1$ and $y_2$, such that $$ P(X \mid Y = y_1) \neq P(X \mid Y = y_2). $$

Number of parameteres of a distribution

Given discrete random variables $X_1, \dots, X_n$ that take $\alpha_1, \dots, \alpha_n$ values, respectively, number of parameters to represent:

• $P(X_1, X_2, \dots, X_n) \rightarrow \alpha_1 \alpha_2 \dots \alpha_n$.
• $P(X_1 \mid X_2, \dots, X_n) \rightarrow (\alpha_1 - 1) \alpha_2 \dots \alpha_n$.
  explain