What is the Distribution of the Sum of Two Standard Brownian Motions?

[BrownianMan]

B(t) is a standard Brownian Motion. u and v are both => 0. What is the distribution of B(u) + B(v)?

The mean is 0.

For the variance I get Var(B(u)+B(v)) = u+v. Is this right?

 

[Ray Vickson]

B(t) is a standard Brownian Motion. u and v are both => 0. What is the distribution of B(u) + B(v)?

The mean is 0.

For the variance I get Var(B(u)+B(v)) = u+v. Is this right?

How did you get this?

 

[BrownianMan]

Aren't B(u) and B(v) independent? If so, then the variance of their sum should be the sum of their variance.

 

[Ray Vickson]

Aren't B(u) and B(v) independent? If so, then the variance of their sum should be the sum of their variance.

Is $\text{Var}( B(1) + B(1))$ equal to 2? Is $2^2$ equal to 2?