by X is 
PROPOSITION 1.7.– Let X be a random variable on (Ω,
This technical result will be useful in certain demonstrations further on in the text. In general, if it is known that Y is σ(X)-measurable, we cannot (and do not need to) make explicit the function f. Reciprocally, if Y can be written as a measurable function of X, it automatically follows that Y is σ(X)-measurable.
EXAMPLE 1.20.– A die is rolled 2 times. This experiment is modeled by Ω = {1, 2, 3, 4, 5, 6}2 endowed with the σ-algebra of its subsets and the uniform distribution. Consider the mappings X1, X2 and Y from Ω onto ℝ defined by
thus, Xi is the result of the ith roll and Y is the parity indicator of the first roll. Therefore, thus, Y is σ(X1)-measurable. On the other hand, Y cannot be written as a function of X2.
The σ-algebra generated by X represents all the events that can be observed by drawing X. It represents the information revealed by X.
DEFINITION 1.14.– Let (Ω,
– Let X and Y be two random variables on (Ω, , ℙ) taking values in (E1, ε1) and (E2, ε2). Then, X and Y are said to be independent if the σ-algebras σ(X) and σ(Y) are independent.
– 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.
PROPOSITION 1.8.– If X and Y are two integrable and independent random variables, then their product XY is integrable and
1.2.4. Random vectors
We will now more closely study random variables taking values in ℝd, 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,
is such that for any i ∈ {1, ..., d}, Xi is a real random variable.
DEFINITION 1.15.– A random vector is said to be discrete if each of its components, Xi, is a discrete random variable.
DEFINITION 1.16.– Let
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 :
