always signifies mutual independence and not pairwise independence.
1.2.2. Random variables
Let us now recall the definition of a generic random variable, and then the specific case of discrete random variables.
DEFINITION 1.9.– Let (Ω,
In the specific case where E = ℝ and = ε =
EXAMPLE 1.12.– Let us return to the experiment where a six-sided die is rolled, where the set of possible outcomes is Ω = {1, 2, 3, 4, 5, 6}, which is endowed with the uniform probability. Consider the following game:
– if the result is even, you win 10 ;
– if the result is odd, you win 20 .
This game can be modeled using the random variable defined by:
This mapping is a random variable, since for any B ∈
and all these events are in
DEFINITION 1.10.– The distribution of a random variable X defined on (Ω,
The distribution of X is a probability distribution on (E, ε); it is also called the image distribution of ℙ by X.
DEFINITION 1.11.– A random real variable is discrete if X(Ω) is at most countable. In other words, if X(Ω) = xi, i ∈ I, where I ⊂ ℕ . In this case, the probability distribution of X is characterized by the family
EXAMPLE 1.13.– Uniform distribution: Let
It is then said that X follows a uniform distribution on {x1, ..., xN }.
EXAMPLE 1.14.– The Bernoulli distribution: Let p ∈ [0, 1]. Let X be a random variable on (Ω,
It is then said that X follows a Bernoulli distribution with parameter p, and we write X ∼
The Bernoulli distribution models random experiments with two possible outcomes: success, with probability p, and failure, with probability 1 – p. This is the case in the following game. A coin is tossed N times. This experiment is modeled by Ω = {T, H}N, endowed with the σ-algebra of its subsets and the uniform distribution. For 1 ≤ n ≤ N, the mappings Xn from Ω onto ℝ are considered, defined by
the number of tails at the nth toss. Thus, Xn, 1 ≤ n ≤ N, are random real variables in the Bernoulli distribution with parameter 1/2 if the coin is balanced.
EXAMPLE 1.15.– Binomial distribution: Let p ∈ [0, 1],
It is then said that X follows a binomial distribution with parameters N and p, and we write X ∼
If the Bernoulli experiment with probability of success p is repeated N times, independently, then the binomial distribution is the distribution of the random variable containing the number of successes at the end of the N repetitions of the experiment.
