Independence
The definition that was given earlier for independence of two random variables holds for both discrete and continuous random variables; it is repeated here.
Definition
Two random variables, \(X\) and \(Y\), are independent if all events about the value of \(X\) are independent of all events about the value of \(Y\).
Independence of continuous random variables is usually deduced from the way that the variables are measured rather than from mathematical calculations. For example,
Characterisation of independence
For values of independent continuous random variables, \(X\) and \(Y\), that are very close to \(x\) and \(y\) — no more than \(\delta x\) and \(\delta y\) above them,
\[ \begin{align} P(x \lt X \lt x+\delta x &\textbf{ and } y \lt Y \lt y+\delta y) \\ &=\;\; P(x \lt X \lt x+\delta x) \times P(y \lt Y \lt y+\delta y) \end{align} \]Using an earlier approximation,
\[ P(x \lt X \lt x+\delta x \textbf{ and } y \lt Y \lt y+\delta y) \;\;\approx\;\; f_X(x)\;f_Y(y) \times \delta x \; \delta y \]where \(f_X(x)\) and \(f_Y(y)\) are the probability density functions of \(X\) and \(Y\). Probabilities around \(x\) and \(y\) are therefore characterised by the product of the pdfs of the variables,
\[ P(X \approx x \textbf{ and } Y \approx y) \;\; \propto \;\; f_X(x)\;f_Y(y) \]This is closely related to the corresponding result for two independent discrete random variables,
\[ P(X=x \textbf{ and } Y=y) \;\;=\;\; p_X(x) \times p_Y(y) \]Random samples
The definition of a random sample that was given earlier also holds for continuous distributions — a collection of \(n\) independent identically distributed random variables from the same distribution is called a random sample.
Extending our earlier characterisation of independence of two continuous random variables,
\[ P(X_1 \approx x_1, X_2 \approx x_2, ..., X_n \approx x_n) \;\; \propto \;\; \prod_{i=1}^n f(x_i) \]This is again closely related to the corresponding formula for a random sample from a discrete distribution
\[ P(X_1 = x_1, X_2 = x_2, ..., X_n = x_n) \;\; = \;\; \prod_{i=1}^n p(x_i) \]