# DISCRETE RANDOM VARIABLES

3.2.1 Univariate Random Variables

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 |

(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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