Difference Between mutual independent and pairwise independent They don't influence each other doesn't mean there's no effect when combined
A ⊥ B A \perp B A ⊥ B P ( A , B ) = P ( A ) P ( B ) P(A, B) = P(A)P(B) P ( A , B ) = P ( A ) P ( B ) Product rule which involves Bayes Theorem P ( X , Y ) = P ( X ) P ( Y ∣ X ) = P ( Y ) P ( X ∣ Y ) = P ( X ∩ Y ) P(X, Y) = P(X)P(Y|X) = P(Y)P(X|Y) = P(X \cap Y) P ( X , Y ) = P ( X ) P ( Y ∣ X ) = P ( Y ) P ( X ∣ Y ) = P ( X ∩ Y ) Joint probability of X and Y is simply understood as choosing X first and choose Y given X. Confusing part is that regardless of order, reverse probability is expressed as same form.
After that we can decide
Pairwise Independence or not
P ( X ) = P ( X ∣ Y ) , P ( Y ) = P ( Y ∣ X ) P(X) = P(X|Y), P(Y) = P(Y|X) P ( X ) = P ( X ∣ Y ) , P ( Y ) = P ( Y ∣ X ) Property E [ X Y ] = ∑ ∑ x y ⋅ P ( X , Y ) = ∑ x P ( X ) ∑ y P ( Y ) = E [ X ] E [ Y ] E[XY] = \sum\sum xy \cdot P(X, Y) = \sum xP(X) \sum yP(Y) = E[X]E[Y] E [ X Y ] = ∑∑ x y ⋅ P ( X , Y ) = ∑ x P ( X ) ∑ y P ( Y ) = E [ X ] E [ Y ] If X and Y are independent
V ( X ± Y ) = V ( X ) + V ( Y ) V(X\pm Y) = V(X) + V(Y) V ( X ± Y ) = V ( X ) + V ( Y ) C o v ( X , Y ) = C o r r ( X , Y ) = 0 Cov(X, Y) = Corr(X,Y) = 0 C o v ( X , Y ) = C orr ( X , Y ) = 0
