DISCRETE RANDOM VARIABLES

3.2.1 Univariate Random Variables

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The following are examples of several random variables defined over a given sample space.

Note that Xj can hardly be distinguished from the sample space itself. It indicates the little difference there is between Definition 3.1.1 and Defini­tion 3.1.2. The arrows indicate mappings from the sample space to the random variables. Note that the probability distribution of X2 can be derived from the sample space: P(X2 = 1) = У2 and P(X2 = 0) = У2.

EXAMPLE 3.2.2 Experiment: Tossing a fair coin twice.

Probability

і

4

1

4

1

4

Sample space HH

HT

TH

TT

Подпись:Подпись:image024x2

Подпись: (X,Y)(1Д) (1,0) (0,1) (0,0)

In almost all our problems involving random variables, we can forget about the original sample space and pay attention only to what values a random variable takes with what probabilities. We specialize Definition

3.1.1 to the case of a discrete random variable as follows:

definition 3.2.1 A discrete random variable is a variable that takes a countable number of real numbers with certain probabilities.

The probability distribution of a discrete random variable is completely characterized by the equation P(X = *,•) = pis і = 1, 2, … , n. It means the random variable X takes value хг with probability рг. We must, of course, have = 1; n may be °o in some cases. It is customary to

denote a random variable by a capital letter and the values it takes by lowercase letters.

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