2.3. Random variables

In many cases concerning the outcomes of random experiments, random events can directly be represented by numerical values, e.g. when rolling a dice. Such a representation is very convenient even in cases where the outcomes of random experiments are non-numerical. This can be done by defining a correspondence between the outcomes of any type of random experiment to numerical values. Such a correspondence (mathematically a function) is called a random variable (RV).

Definition: A random variable is a function from the sample space Ω to the set of real numbers R. The range of a random variable is the set of values the random variable can take. We denote random variables by capital letters (X, Y, Z) and their values by lowercase letters (x, y, z). Notation: X: ω → X(ω) defined for all ω ε Ω.

Exercise 2.3.1 For three tosses of a fair coin, the sample space is Ω = {{{H, H, H}, {H, H, T}}, {{H, T, H}, {H, T, T}}, {{T, H, H}, {T, H, T}}, {{T, T, H}, {T, T, T}}} with H denoting head and T denoting tail. One of the many possible examples for a random variable defined on this sample space would be X = total number of heads obtained. The range in this case would be the set {0,1,2,3} and we have X({H, H, H}) = 3; X({H, T, H}) = 2; X({T, T, T}) = 0, and so on.

The concept of random variables unifies the treatment of random phenomena with different types of outcomes and simplifies their notation. In addition, the use of random variables facilitates the treatment of combinations of random phenomena. For example, if X and Y are random variables defined on a sample space, any real-valued function of X and Y, is also a random variable (if the sample space is countable). The term random variable is somewhat unfortunate, since in contrast to other mathematical variables, a random variable cannot be assigned a single value. Instead it is described by the set of all possible values of X (the range of the RV) together with the probability to obtain each of the values. Loosely speaking, a random variable has a probability distribution. Depending on the type of the range of a random variable, random variables come in two flavors. If the range is finite or countably infinite the random variable is said to be discrete ; if it is an interval or the union of intervals, it is said to continuous.

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.............. 2.3. Random variables

.............. 2.3.1. Distribution functions

.............. 2.3.2. Representative measures of probability distributions