Two events are independent if and only if

If $X$ and $Y$ are independent:

Joint probability is equal to the product of the Marginal Distributions

Joint cumulative probability is equal to the product of the Marginal Distributions

If $\mathrm{\exists }{x}_0,y_0$ such that $f_{X,Y}(x_0,y_0)\ne f_X(x_0)f_Y(y_0)$ then $X$ and $Y$ are not independent

Discrete

For $n$ discrete random variables $X_1,X_2,\dots ,X_n$ with a joint probability distribution $f(x_1,x_2,\dots ,x_n)$ and marginals distributions $f_{X_i}(x_i)$:

if and only if the $n$ random variables are independent

Three of more random variables can still be pairwise independent without being completely independent among all of them

Continuous

For $n$ continuous random variables $X_1,X_2,\dots ,X_n$ with a joint probability densities $f(x_1,x_2,\dots ,x_n)$ and marginals densities $f_{X_i}(x_i)$:

if and only if the $n$ random variables are independent

The cdfs can be used instead to determine the dependency of the variables

Measures

• $E(XY)=E(X)E(Y)$
• $Var(X+Y)=var(X)+var(Y)$
• $cov(X,Y)={\sigma }_{XY}=0$