Let $B(t)$ denote the standard Brownian motion and let $X(t)$ denote a Brownian motion with $X(0)=0$, drift $0$ and variance $9$.
Find the distribution of $aB(s)+bB(t)$, where $a,b,s,t$ are real numbers and $0
Find $\Bbb P(X(2) -2X(3) ≤ 4)$
Let $B(t)$ denote the standard Brownian motion and let $X(t)$ denote a Brownian motion with $X(0)=0$, drift $0$ and variance $9$.
Find the distribution of $aB(s)+bB(t)$, where $a,b,s,t$ are real numbers and $0
Find $\Bbb P(X(2) -2X(3) ≤ 4)$
We write $aB(s)+bB(t)=b(B(t)-B(s))+bB(s)+aB(s)=b(B(t)-B(s))+(a+b)B(s).$ As $B(t)-B(s)$ and $B(s)$ are independent, and normally distributed of respective laws $N(0,t-s)$, and $N(0,s)$ (definition of Brownian motion), $aB(s)+bB(t)$ is normally distributed, of mean $0$ and variance $b^2(t-s)+(a+b)^2s=b^2t-sb^2+a^2s+2abs+b^2s=b^2t+a^2s+2abs.$
The second question almost follows from the first one.