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 – Any family (Xi)i∈I of random variables is independent if the σ-algebras σ(Xi) are independent.

 – Let be a sub-σ-algebra of , and let X be a random variable. Then, X is said to be independent of if σ(X) is independent of or, in other words, and are independent.

We will now more closely study random variables taking values in ℝ^d, with d ≥ 2. This concept has already been defined in Definition 1.9. We will now look at the relations between the random vector and its coordinates. When d = 2, we then speak of a random couple.

i

i

The conjoint distribution (or joint distribution or, simply, the distribution) of X is given by the family

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and

The concept of joint distributions and marginal distributions can naturally be extended to vectors with dimension larger than 2.

EXAMPLE 1.21.– A coin is tossed 3 times, and the result is noted. The universe of possible outcomes is Ω = {T, H}³. Let X denote the total number of tails obtained and Y denote the number of tails obtained at the first toss. Then,