I understand that intuitively two R.V.s are independent if knowing one does not change the probability of the other one. This intuitive definition can also be shown mathematically. However, I am a bit confused by the following example.
Let's say we know $X_1$ and $X_2$ are two Gaussian random variables with unknown parameter $\theta$, i.e., $X_1, X_2 \sim \mathcal{N}(\theta,1)$. Now if we know either $X_1$ or $X_2$, then we can infer something about $\theta$ and hence knowing one changes the probability of the other. Consequently, $X_1$ and $X_2$ are dependent by the intuitive definition. I know something is wrong with this logic, otherwise we cannot have independent and identically distributed (i.i.d) random variables.
Could someone help me on this?
Thanks.