I have the following question in mind which I wanted to answer: what is the measure of the set which will be covered by a standard Brownian motion $B(t)$ in a time $t$? Call this random variable $M(t)$. I want to find $P(M(t)\leq x)$. Call the probability density of this $h(x)$.
What I thought was the following: Denote by $f(x)$ and $g(x)$ the probability densities of respectively $P(-\min_{0\leq s\leq t}B(s)\leq x)$ and $P(\max_{0\leq s\leq t}B(s)\leq x)$. Then we have that $$h(y)=\int_0^{\infty}f(x)g(y-x)dx$$ if the minimum and maximum of Brownian motion are independent. I don't know if this is the case. If someone can help me with this, I would really appreciate.