I've come across an interesting question while studying for my final exams and was looking for some clarification.
The Question
Let X have a uniform Expo(3) distribution and let A be the event that X equals a natural number. Let B be the event that the value of X is less than 0.5.
Are A and B mutually exclusive? Are A and B (stochastically) independent?
The Attempt at a Solution
My initial intuition was that A and B were mutually exclusive because both events could not occur at the same time, and they were hence dependent, as knowing that A occurred told us with certainty that B did not. However, looking at it more closely, X is a continuous distribution, which means A cannot happen (as X can never be equal to a natural number) which would lead me to believe that my intuition is incorrect.
I am seeking clarification on how to incorporate this information into the definitions of mutual exclusion and independence, as P(A) = 0 must be true if X is a continuous distribution. Insight into this problem would be greatly appreciated, as it is something I wish to clear up before my exam.
As a curiosity, what would change if X had a different continuous distribution, such as Uniform(a,b) or Normal(0,1)?
Regards, Kurndles