Probability Basics and Expected Value

Probability is defined as a chance that a random number will have a particular outcome. For example, there is a 50-50 chance (hence 50% probability) that a fair coin will have a heads land up. If you gather all the possible outcomes of a random variable, such as the outcomes of the dice when it is rolled up, it generates a distribution which is the key in Probability space as it gives an insight of all possible values of the Random Variable.

What is a Random Variable: 
Random variable is just a name that can be anything where an analysis needs to be performed. Common examples are, outcome of a toss coin, outcome of a Dice roll, Daily returns of the financial instrument and so on. Once you know the outcomes of the Random variable, you can assign the probabilities attached to it to generate Probability Distribution. It is this probability distribution that helps to forecast future outcomes with certain confidence.

There are 4 main topics that one must consider while working with Probability.
1. CDF - Cumulative Distribution Function - CDF, F(.), of a random variable, X, is defined as:
                F(x) := P(X <= x)
                (Probability of X is less than or equal to x)

2. PMF - Probability Mass Function. p(.) is a Probability Mass function for the random variable, X,
                P(X in A) = Summation (p(x)) for all x in A

3. Expected Value: Expected value of a random variable, X, is given by the sigma of probability weights of the random variable.
                E(X) := Summation (Random value * Probability of Random Value)

4. Variance: Variance of a Random variable, X, is defined as
                Var(X) := E[(X - E[X]^2)]
                             = E[X^2] - E[X]^2
Variance is also known as Mean of the Squares - Square of the Mean.

Real World Application: 
What is the expected value of a fair dice roll?
As we know a fair dice will have 6 faces, numbered from 1 through 6 with each face with equal probability of 0.1667 or 16% chance of each number showing up.

Expected value then becomes:
0.1667 *(1) + 0.1667 *(2) + 0.1667 *(3) + 0.1667 *(4) + 0.1667 *(5) + 0.1667 *(6) = 3.5

Therefore one can say that the average value of a dice outcome is 3.5 or there is a 50-50 chance of beating the average value in which case 50% of the times, a dice can have a value higher than Expected value (4,5,6) or 50% chance that it can have lower than expected values of 3.5 (1,2,3).

Simple stuff - isn't it?

Nitin