independence

Definition

For events $A$ and $B$ to be independent,

independence of random variables

Suppose we have random variables $X_1,…,X_k$. We say $X_i$ and $X_j$ are independent if, for all possible values $v_i,v_j$,

(in this case event $A$ is when $X_i = v_i$, and $B$ when $X_j = v_j$)

cdf and pdf

$X$ and $Y$ are independent iff pdf adheres to

$X$ and $Y$ independent iff cdf adheres to

conditional probability when independent

When events $A$ and $B$ are independent, then,

(this follows from definition of conditional probability and the definition of independence given above)

independent r.v.'s are uncorrelated

When two random variables are independent, then they are uncorrelated (see correlation coefficient). However, the converse does not necessarily hold.

See also

• types of independence
• pairwise independence
• mutual independence

References:

1. https://www.mathcounterexamples.net/mean-independent-and-correlated-variables/