A random variable is a variable whose value, which cannot be known beforehand, is determined by the outcome of an experiment.
The probability of its occurrence can be predicted.
Types of Random Variables
Discrete random variables are often the result of a counting process and are often whole numbers. e.g. Number of heads when tossing a coin.
Continuous random variables are often the result of measurement and are usually from the set of real numbers e.g. Heights of basketball players.
Notation
P (X = x) means the probability that the random variable X has the value of x.
e.g. P( D = 6) could mean the probability that when a die is rolled a "6" has been scored.
Probability Distribution
As a table
A probability distribution or probability function is a table which shows the probabilities associated with each value of a random variable.
A die is thrown and the results are recorded. Let the discrete random variable X =the number shown on the die.
The probability distribution looks like this.
X 
1

2

3

4

5

6

P(X = x) 
Note that the sum of the probabilities in the probability distribution of a discrete random variablealways add up to 1.
i.e. Σ P(X = x) = 1
In the above case, Σ P(X = x) = + + + + + = 1
As a formula
A probability function can also be written as a formula where each value of x is substituted into the formula to give the probability that the value x will occur.
P(X = x) = where x =1, 2 ,3, 4 is the probability function for a circular spinner used in a game of chance. When x = 1, P(x = 1) = = 0.1 etc. 
This gives a probability distribution of
X 
1

2

3

4

P(X = x) 
0.1

0.2

0.3

0.4

ΣP(X = x) = 0.1 + 0.2 + 0.3 + 0.4 = 1
As a graph
A probability distribution can also be shown on a vertical line graph.
For the probability distribution of the spinner experiment above: