sorry
if X(t) = -1 for 0 <= t <= 1
X(t) = 1 for 1 < t <= 2
X(t) = 2 for 2 < t <= 3
then the ito integral int_0^3 X(t)dB(t) = 2B(3) - B(2) - 2B(1)
that i understand. But why does 2B(3) - B(2) - 2B(1) have a distribution Normal(0,6) or if put as sums N(0,1) + N(0,1) + N(0,4).
I know from the definition of wiener process that Bt - Bs = N(0,t-s) for 0<s<t
I have only a casual knowledge of Brownian motion, but I think it goes like this. The integral is
int_0^3 X(t)dB(t) = 2(B(3)-B(2)) + (B(2)-B(1)) - (B(1)-B(0)).
That (B(t) - B(t-1)) ~ N(0,1) for t = 1,2,3 follows from (B(t) - B(s)) ~ N(0,t-s) when s = t-1 and B(0) = 0. Next, after the multiplication by 2, 2(B(3)-B(2)) ~ N(0,2^2) = N(0,4). Finally, use the sum of independent normal variables has a normal distribution with variance equal to the sum of the variances to get
2(B(3)-B(2)) + (B(2)-B(1)) - (B(1)-B(0)) ~ N(0,4+1+1) = N(0,6).