Independence of continuous random Variables

Two discrete random variables X and Y with support \(T_X = {x_1, x_2, \dots, x_k, \dots }\) and \(T_Y = {y_1, y_2, \dots, y_k, \dots }\) are called (stochastically) independent if for every x_i T_X and every Y_i T_Y:

\[P(X = x_i, Y = y_i) = P(X = x_i) * P(Y= y_i\]

• Hence: X and Y are independent if all pairs of events \({X = x_i}\) and \({Y = y_i}\) are independent

More generally:
If the discrete random variables X and Y are independent, then for any sets \(M_1, M_2 \in R\) the events \({X \in M_1} and {Y \in M_2}\) are also independent, i.e. it holds that \(P(X \in M_1, Y \in M_2) = P(X \in M_1) * P(Y \in M_2)\)

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