Probability of uniforms spaces

Question

Let X be a random variable on probability space (Ω,P), Suppose X~U({1,2,3}), does this mean the space (Ω,P) is uniform space. I tried to come up with counter example and did not work out but i still think this statement isn't right.

Answer 1

You are correct that the statement isn't right. Here is a concrete counterexample with a bit of R code to illustrate it.

A standard 6-sided die has 3 pairs: (1,6), (2,5), (3,4) where each number in the pair is on the opposite side of the other. Suppose that such a die is biased so that each pair is equally likely but that in a pair the larger of the two numbers is twice as likely as the smaller. For example, 6 is twice as likely as 1. This is easily seen to imply that the numbers 1,2,3 appear with probability 1/9 and the numbers 4,5,6 appear with probability 2/9.

You can simulate 1000 rolls like this:

rolls <- sample(1:6,1000,replace = TRUE, prob = c(1/9,1/9,1/9,2/9,2/9,2/9))

Here is a display created by making a barplot of the tabulation of the results:

Confirming the obvious fact that the distribution is not uniform.

We can define X on it as the function which indicates what pair a rolls is in (so that 1,6 are in the first pair, 2,5 in the second, 3,4 in the third):

X = function(x){min(x,7-x)}

and then:

barplot(table(sapply(rolls,X)))

leading to:

which confirms the obvious fact that X is uniform.