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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.

PROPOSITION 1.9.– Let X be a real random vector on the probability space (Ω, , ℙ), taking values in ℝ^d. Then,

is such that for any i ∈ {1, ..., d}, X_i is a real random variable.

DEFINITION 1.15.– A random vector is said to be discrete if each of its components, X_i, is a discrete random variable.

DEFINITION 1.16.– Let be a discrete random couple such that

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

The marginal distributions of X are the distributions of X₁ and X₂. These distributions may be derived from the conjoint distribution of X through:

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,

The couple (X, Y) is, therefore, a random vector (referred to here as a “random couple”), with joint distribution defined by

for any (i, j) X(Ω) × Y (Ω), which makes it possible to derive the distributions of X and Y (called the marginal distributions of the couple (X, Y )):

Distribution of X:

Distribution of Y :