SMSTC (2007/08) Probability

                  www.smstc.ac.uk

Contents

1 Random variables 1–1 2 Joint distributions and independence 2–1 3 Important special distributions 3–1

SMSTC (2007/08) Probability

1: Random variables

Takis Konstantopoulos, Heriot-Watt University^{a}

                  www.smstc.ac.uk

A random variable is a measurable function X between two measurable spaces, an “abstract” one (Ω, ℱ) and a “concrete” one (S, 𝒮). In other words, we require that the inverse image by X of each element of 𝒮 be an element of ℱ. We denote this, implicitly, by the symbol

X : (Ω, ℱ) → (S, 𝒮).

The terms in quotes have nothing to do with Mathematics but, rather, with our human interpretation of it. For example, we want, most of the time, to leave (Ω, ℱ) unspecified but (S, 𝒮) can be, for example, (ℝ, ℬ), where ℬ is the class of Borel sets on ℝ. In such a case we call X ONE random variable, omiting the numeral ONE when not necessary. As another example, let (S, 𝒮) = (ℝ^{2}, ℬ^{2}), where ℬ^{2} are the Borel sets on ℝ^{2}. In this case we refer to X as TWO random variables, because we may, by choosing Cartesian coördinates on the ℝ^{2}, represent X by (X_{1}, X_{2}), where X_{1} is one random variable and X_{2} is also one random variable. More generally, we may let (S, 𝒮) = (ℝ^{d}, ℬ^{d}), and then we refer to X a random (d-dimensional) vector.

The real fun of course starts when (S, 𝒮) is chosen to be a large (but concrete) space, for example the space of real-valued continuous functions on the real line. In this case, 𝒮 is usually chosen to be the minimal σ-algebra containing the open sets of continuous functions, open with respect to the topology of uniform convergence on compacta. In such a case, X is called random function or stochastic process (stochastic being a synonym of random and process being a synonym of function).

Of course, the concept of “random variable” has nothing to do with randomness. The adjective “random” is attached because, in addition to the above, we also consider some probability (measure) P on (Ω, ℱ) . As a mathematical definition, the concept of random variable is purely functional. So let us do that: let P be a probability on (Ω, ℱ). The distribution of the random variable X is a probability P_{X} on (S, 𝒮) which is induced, in the most natural fashion, by X:

P_{X}(B) = P(X^{−1}(B)), B ∈ 𝒮 .

1–1

SMSTC (2007/08) Probability

2: Joint distributions and independence

Takis Konstantopoulos, Heriot-Watt University^{a}

        www.smstc.ac.uk

2–1

SMSTC (2007/08) Probability

3: Important special distributions

Takis Konstantopoulos, Heriot-Watt University^{a}

        www.smstc.ac.uk

3–1