A variable is considered as random when:

• its values are numerical,
• it is defined based on a sample space, and corresponds to an event,
• it has an associated probability.

These characteristics are illustrated in Fig. 1.

Fig. 1 Relation between random variable, sample space and event

By convention, random variable is symbolized by upper case character (`X,\ Y,\ A,\ B,\ ...\ `), whereas its value is symbolized by lower case one (`x,\ y,\ a,\ b,\ ...\ `).

The following examples will clarify these notions.

Example 1 : Rolling two dice A and B. Outcome is defined as the upper faces and symbolized as `(ab)`, which `a` is the upper face of die A and `b` is the upper face of die B. So sample space consists of 36 elements:

     S = { (11), (12), (13), (14), (15), (16),
           (21), (22), (23), (24), (25), (26),
           (31), (32), (33), (34), (35), (36),
           (41), (42), (43), (44), (45), (46),
           (51), (52), (53), (54), (55), (56),
           (61), (62), (63), (64), (65), (66) }

Define `X` as the sum of `a` and `b` : `X=a+b`

`X` is a random variable because:

• its values are numbers and in the range of 2 to 12,
• associate with value `x` (of `X`) is the event E_`x`,
• associate with value `x` is the probability `P(X = x)`. Example:
  • `x=4` : E₄ = { (13), (22), (31) } ; `P(X=4)=P("E"_4)=3//36`
  • `x=9` : E₉ = { (36), (45), (54), (63) } ; `P(X=9)=P("E"_9)=4//36`

Example 2 : With the sample space in Example 1, we can define `Y` as the difference of `a` and `b`: `Y=a-b`. `Y` is also a random variable with:

• value `y` of `Y` is in the range -5 (16) to 5 (61),
• F_`y` is the event associate with `y`.
• `y=3` : F₃ = { (41), (52), (63) } ; `P(Y=3)=P("F"_3)=3//36`

Example 3 : Tossing two coins 1 and 2. If the upper face of a coin has picture or icon on it, the outcome of that coin is symbolized as H (head). On the contrary (the upper face has number on it) the outcome is symbolized as T (tail). Outcome of this experiment is defined as the upper faces of two coin and symbolized as AB, which A is the upper face of coin 1 and B is the upper face of coin 2. So sample space of this statistical experiment consists of 4 elements:

    S = { HH, HT, TT, TH }

Define `X` as number of H. X is a random variable because:

• three values of `X` are numerical : 0, 1, and 2,
• associated with `x` = 0 is E₀ = { TT } ; `P("E"_0)=1//4`,
• associated with `x` = 1 is E₁ = { HT, TH } ; `P("E"_1)=2//4`,
• associated with `x` = 2 is E₂ = { HH } ; `P("E"_2)=1//4`.

• from one sample space, we can define many random variables,
• values of random variable are not necessary belonging to sample space,
• values of random variable are not necessary to have the same data type with elements of sample space,
• we can define a random variable even when elements of sample space are not numerical,

Depend on values of a random variable, it can be discrete or continuous.

• A random variable is discrete if its number of values is definite or countable (“countable” means that we can associate values of variable with a counting procedure). For example number of defect in sample.
• A random variable is continuous if its number of values is indefinite, cannot count, or we cannot associate its values with any counting procedure. For example the weight of sample.

Probability distribution is the association between a value `x` of random variable and the probability of corresponding event. The relation between these notions is illustrated in Fig. 2

Fig. 2 The relation between random variable, event and probability distribution.

Probability distribution can be presented by:

• formula
• table
• chart

All these forms will be discussed in "Applied Statistics and Design of Experiments".