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Probability, Fourth Edition by Rick Durrett

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2Laws of Large Numbers

2.1 Independence

Measure theory ends and probability begins with the definition of independence. We begin with what we hope is a familiar definition and then work our way up to a definition that is appropriate for our current setting.

Two events A and B are independent if P(A ∩ B) = P(A)P(B).

Two random variables X and Y are independent if for all C, D ∈ Images,

Images

that is, the events A = {X ∈ C} and B = {Y ∈ D} are independent.

Two σ-fields and are independent if for all A ∈ and B ∈ the events A and B are independent.

As the ...

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