Probability Frequency Distribution

A probability distribution Pr{} on a sample space S is a mapping from events of S to real numbers satisfyingly the following probability axioms:

• Pr{A} >= 0 for any event A
• Pr{S} = 1
• Pr{A U B} = Pr{A} + Pr{B} for any two mutually exclusive events A and B. This can be applied more generally as the sum of mutually exclusive events.

The PFD is the collection of the probabilities for each possible outcome.

• For example: The probability of rolling 2 dices and that their result is 7 is 1/6 since there are 6 favorable outcomes out of 36 possible ones.

	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

• For example: What is the expected value of rolling two dice and summing the result?
  • E(X) = 2 * P(2) + 3 * P(3) + … + 12 * P (12) = 7

SUM	FREQ	PROB
2	1	1/36
3	2	2/36
4	..	..
5	..	..
6	5	5/36
7	6	6/36
...
8	5	5/36
9	4	4/36
...
12	1	1/36

Note how different intervals have different probabilities of being realized.