Let $\{B_t:t\ge0\}$ be a standard brownian process. What is the expectation of $E(\min_{1\le s\le 2}B_S)$?
I think the problem is i am not sure how $\min_{1\le s\le 2}B_S$ is distributed. I try that $\min_{1\le s\le 2}B_S=X_s$, $P(X_s\le x)=P(x\le B_s)=1-P(B_s\le x) \text{for}\ s\in[1,2]$ but i am not sure how to proceed or is there any other easier method to find the expectation?