Random Variables

At the level of an intermediate textbook, a random variable is defined as a real-valued function over a sample space. But a sample space is not defined precisely, and once a random variable is defined the definition is quickly forgotten and a random variable becomes identified with its probability dis­tribution. This treatment is perfectly satisfactory for most practical applica­tions, but certain advanced theorems can be proved more easily by using the fact that a random variable is indeed a function.

We shall first define a sample space and a probability space. In concrete

terms, a sample space may be regarded as the set of all the possible outcomes of an experiment. Thus, in the experiment of throwing a die, the six faces of the die constitute the sample space; and in the experiment of measuring the height of a randomly chosen student, the set of positive real numbers can be chosen as the sample space. As in the first example, a sample space may be a set of objects other than numbers. A subset of a sample space may be called an event. Thus we speak of the event of an ace turning up or the event of an even number showing in the throw of a die. With each event we associate a real number between 0 and 1 called the probability of the event. When we think of a sample space, we often think of the other two concepts as well: the collection of its subsets (events) and the probabilities attached to the events. The term proba­bility space refers to all three concepts collectively. We shall develop an ab­stract definition of a probability space in that collective sense.

Given an abstract sample space Q, we want to define the collection A of subsets of Cl that possess certain desired properties.

Definition 3.1.1. The collection A of subsets of £2 is called a o-algebra if it satisfies the properties:

(i) Cl E A. __ __

(ii) EE A=* E E A. (E refers to the complement of E with respect to Q.)

(iii) EjEA, ;’= 1, 2,. . .=> u;_, Ej E A.

Given a ff-algebra, we shall define over it a real-valued set function satisfy­ing certain properties.

Definition 3.1.2. A probability measure, denoted by P( •), is a real-valued set function that is defined over a (7-algebra A and satisfies the properties:

(i) ЕЕА=*Р(Е)Ш0.

(ii) P(Cl)= 1.

(iii) If {Ej) is a countable collection of disjoint sets in A, then

image173

A probability space and a random variable are defined as follows:

Definition 3.1.3. Given a sample space Cl, a (7-algebra A associated with Q, and a probability measure P{ •) defined over A, we call the triplet (Cl, A, P) a probability space.3

Definition 3.1.4. A random variable on (П, A, P) is a real-valued func­tion4 defined over a sample space Q, denoted by X(co) for соє Q, such that for any real number x,

(coX(co) <x) Є A.

Let us consider two examples of probability space and random variables defined over them. ,

Example 3.1.1. In the sample space consisting of the six faces of a die, all the possible subsets (including the whole space and the null set) constitute a <7-algebra. A probability measure can be defined, for example, by assiging 1/6 to each face and extending probabilities to the other subsets according to the rules given by Definition 3.1.2. An example of a random variable defined over this space is a mapping of the even-numbered faces to one and the odd-num­bered faces to zero.

Example 3.1.2. Let a sample space be the closed interval [0, 1]. Consider the smallest ст-algebra containing all the open sets in the interval. Such a сг-algebra is called the collection of Borel sets or a Borel field. This a-algebra can be shown to contain all the countable unions and intersections of open and closed sets. A probability measure of a Borel set can be defined, for example, by assigning to every interval (open, closed, or half-open and half – closed) its length and extending the probabilities to the other Borel sets ac­cording to the rules set forth in Definition 3.1.2. Such a measure is called Lebesgue measure.5 In Figure 3.1 three random variables, X, Y, and Z, each of which takes the value 1 or 0 with probability £, are depicted over this probabil-

1 1 1

XU)

YU)

ZU)

0

•10;

1

0 –

І 2

:

Г <

ї 1

w to a)

Figure 3.1 Discrete random variables defined over [0, 1] with Lebesgue measure

ity space. Note that Z is independent of either X or Y, whereas X and Y are not independent (in fact XY = 0). A continuous random variable X(qj) with the standard normal distribution can be defined over the same probability space by X— Ф-‘(&>), where Ф is the standard normal distribution function and 1” denotes the inverse function.

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