I am trying to learn Brownian motion on my own and attempting some exercises.
Let ${X_t}$ be a standard 1D Brownian motion and define
$$T=\inf \{t:X_t=1 \ or \ -3 \}$$
The question is: show that $T$ and $X_T$ are not independent.
Intuitively, this makes sense because $1$ is closer to the origin than $-3$, so if $T$ is very small, the Brownian motion probably has not displaced much so probably $X_T =1$. As $T$ gets larger, the probability starts shifting towards $X_T=-3$.
However, I can't seem to prove the absence of independence mathematically. Although this is one possible approach (albeit maybe unnecessarily overkill), is it possible to find $P(X_T = 1 | T=t)$ and directly show that it depends on $t$? As a bonus, $P(X_T = 1 | T=t)$ would be interesting to look at.
Source: Introduction to Stochastic Processes, by Lawler, Problem 8.13(b).