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Random variables are said to be independent ($\perp$ perp) if the product of the probabilities is equals to the probability of the events $X_1, ..., X_n$.

\[ \mathbb{P}(X_1 = x_1,\ \ldots,\ X_n = x_n) = \prod_{i=1}^n \mathbb{P}(X_i = x_i) \]

Remember that the , is read as and, so you could also write

\[ \mathbb{P}(X_1 = x_1\ \cap\ \ldots\ \cap\ X_n = x_n) = \prod_{i=1}^n \mathbb{P}(X_i = x_i) \]

Another way is if we have (the previous one is induced from this one)

  • if $\mathbb{P}(A|B) = \mathbb{P}(A)$
  • or $\mathbb{P}(B|A) = \mathbb{P}(B)$