PROBABILITY DISTRIBUTION @Amit Sir Please help

[Mauli]

Suppose I tell you that I have randomly selected a positive integer as follows. I have taken the numbers {1,2,3..} and chosen from a probability distribution such that every positive integer is just as likely as every other positive integer , i.e: 3 is as likely to be chosen as 10 , which is as likely as 381, 384 etc. What is wrong with my claim?

[Ayushya Kaul]

I suppose the problem has to be with the size of the sample space?
Its not finite.

[Amit Goyal]

Hi Mauli,

This is how we define probability:
Probability is a function Pr : Set of all events --> [0, 1]  satisfying the following axioms:
A 1. Pr(S) = 1, Pr({}) = 0
A 2. Pr(E') = 1-Pr(E)   where E' is the compliment of E
A 3. If E(1), E(2), ..... is a finite or infinitely countable collection of mutually disjoint events then
Pr(U E(i)) = ∑ Pr(E(i))

Set  S = {1,2,3, ...} is the set of natural numbers. A uniform probability over the natural numbers would not satisfy all three axioms of probability. By A 1, Pr(S) = 1. By uniform probability and A 3, if
Pr({1}) = p > 0, then Pr(S) = Pr(U {i}) = ∑ Pr({i})  = p+p+..... = ∞ contradicting Pr(S) = 1.
Pr({1}) = 0 , then Pr(S) = Pr(U {i}) = ∑ Pr({i})  = 0 contradicting Pr(S) = 1.

Therefore, we cannot have a uniform probability measure over the set of natural numbers (or any countably infinite set).

[Mauli]

Sir , thanks a lot for replying.
 you have explained this in the most elegant and precise way there is.

and Ayushya thank you for ur prompt reply.

[Ayushya Kaul]

Don't mention it :)