I have a pair of sets:
- $A=\{n\in\mathbb{N}\mid p\cdot n\}$
- $B=\{n\in\mathbb{N}\mid q\cdot n\}$
Where $p$ and $q$ are two different prime numbers.
And the following event definitions:
- $X_n$: $n\in{A}$
- $Y_n$: $n\in{B}$
Does it follow that the events $X_n$ and $Y_n$ are independent for every $n\in\mathbb{N}$ and every pair of different primes?
I believe that $P(X_n\land Y_n)=P(X_n)\cdot P(Y_n)=\frac1p\cdot\frac1q$ for every $n\in\mathbb{N}$ and every pair of different primes, hence the answer is true, but I'm finding it hard to prove this.
Can anyone please help with this?
Thank you!
Since the author doesn't know the probability distribution, I will assume exponential distribution: $$ P(n=k) = \frac122^{-k}, \qquad k=0,1,2... $$
With that assumption, $P(X_n)=\frac{1/2}{1-2^{-p}}$, $P(Y_n)=\frac{1/2}{1-2^{-q}}$, $P(X_n\wedge Y_n)=\frac{1/2}{1-2^{-qp}}$. One can see that these events are not independent.
Can one come up with the distribution, so these events are independent? It is quite another story.