I want to find the following probability :
$\mathbb{P}\{W_1>0, W_5 <0\}$
Where $W_t$ is a Wiener process, so it follows the law $\mathcal{N}(0, t)$.
My question is : can say that $\{W_1>0\}$ and $\{W_5 <0\}$ are uncorrelated ? In that case :
$\mathbb{P}\{W_1>0, W_5 <0\} = \mathbb{P}\{W_1>0\}\mathbb{P}\{W_5 <0\}= (1-\frac{1}{2})\frac{1}{2} = \frac{1}{4}$
As these are both is the cummulative distribution function of a normal law evaluated in $0$.
However, I'm not sure about the assumption I just made. Can someone bring some clarifications ? Thank you.