Kamandirian statistik: Béda antarrépisi

Hal nu sarupa, waktu urang nyebutkeun dua [[variabel acak]] bebas, we intuitively mean that knowing something about the value of one of them does not yield any information about the value of the other. For example, the number appearing on the upward face of a die the first time it is thrown and that appearing the second time are independent.

Here ''A'' ∩∩ ''B'' is the [[intersection (set theory)|intersection]] of ''A'' and ''B'', i.e., it is the event that both events ''A'' and ''B'' occur. Thus we could say:

:Two events ''A'' and ''B'' are '''independent''' [[iff]] P(''A'' ∩∩ ''B'')=P(''A'')P(''B'').

Two random variables ''X'' and ''Y'' are independent iff for any numbers ''a'' and ''b'' the events [''X'' ≤≤ ''a''] and [''Y'' ∈∈ ''b''] are independent events as defined above. Similarly an arbitrary collection of random variables -- possible more than just two of them -- is independent precisely if for any finite collection ''X''₁, ..., ''X''_''n'' and any finite set of numbers ''a''₁, ..., ''a''_''n'', the events [''X''₁ ≤≤ ''a''₁], ..., [''X''_''n'' ≤≤ ''a''_''n''] are independent events as defined above.

The measure-theoretically inclined may prefer to substitute events [''X'' ∈∈ ''A''] for events [''X'' ≤≤ ''a''] in the above definition, where ''A'' is any [[Borel algebra|Borel set]]. That definition is exactly equivalant to the one above when the values of the random variables are [[real number]]s. It has the advantage of working also for complex-valued random variables or for random variables taking values in any [[topological space]].

:E[''X''·· ''Y''] = E[''X''] ·· E[''Y'']

(''Pernyataan'' sabalikna yen lamun dua variabel bebas mangka kovarian-na sarua jeung nol ngarupakeun hal nu teu bener. Tempo [[uncorrelated|taya hubungan]].)

: P[(''X'' in ''A'') & (''Y'' in ''B'') | ''Z'' in ''C''] = P[''X'' in ''A'' | ''Z'' in ''C''] ·· P[''Y'' in ''B'' | ''Z'' in ''C'']

for any Borel subsets ''A'', ''B'' and ''C'' of the real numbers. That is, given ''Z'', the value of ''Y'' does not add any additional information about the value of ''X''.