Probability Distribution

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           The word "Probability" often comes into our day-to-day life referring to, "What is the probability of some random event ?" simply saying what at most chance the event to occur. 

         In the last two blogs "The Axioms - Probability" and "What is the probability you will open this blog ?" we see how probability arises over time. 

     But the question is if the event space contains more than some finite or infinite elements, How will we calculate the probability?

     Here the role of Probability Distribution came into the story. So instead of calculating, we evaluate how the probability is distributed over the region.




      First, take the simplest example, "Tossing a coin", so by the axioms of probability we have here two event spaces called head and tail. The probability of getting the head is 1/2 and the same for the tail. 

      For probability distribution let - < X <  denotes the entire event space and take X=0 corresponding to 'tail' and X=1 corresponding to 'head', so P(X=0)=1/2=P(X=1).

      So exactly what we are doing, we just mapping events in event space (U) to a real number that is X: U -> X(U), called random variable X for a random experiment.

       So two things here play an important role, one is probability density (that is in a certain region or point how much probability is there) and the second is probability distribution (that is how much probability up to a number).

     While doing probability always have to keep in mind the foundational pillars of probability "The Axioms", let us recall the three axioms.

     Let E be a random experiment, if S is the event space and A is any event connected with E. Then,

           1) P(A) ≥ 0

           2) P(S) = 1

          3) If A, B, C, ..... are a finite or infinite sequence of pairwise mutually exclusive events, then P( A + B + C + ...) = P(A) + P(B) + P(C) +.....

    These 3 are like the three legs of a tripod.

    Let's see some more examples," Drawn a random ticket from an urn containing n tickets", So if X is the random variable assume the values 1,2,...,n then P(X=i)=1/n where i = 1,2,...,n.

   Here in above the probability density is the P(X=i)=1/n where i = 1,2,...,n and the probability distribution is P( - < X ≤ i) = i/n.

 Let's see for n=4.



 



       So if we define probability distribution formally, the distribution function of a random variable X is a function of a real variable i, to be denoted by F(i) on (-,) by F(i) = P(- < X ≤ i).

    Try some examples and comment on the answers.

1. Throwing a die (of 6 sides)

2. Drawing a card (total 52 cards)




















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