$W_t$ and $B_t$ are tow independent Brownian motion, where :
$W_t$ ~ $N(0, s^2_1)$,
$B_t$ ~ $N(0, s^2_2)$
$Cov(W_t,B_t)=0$
We know that sum of two Gaussian random variable is also Gaussian.
$$E(1/2(W_t+B_t)) = 1/2(E(W_t+B_t))=0$$ $$Var(1/2(W_t+B_t))=1/4(var(W_t+B_t))=1/4(s^2_1+s^2_2)$$ because $W_t$ and $B_t$ are independent. So:
$1/2(B_t+W_t)$ ~ $N(0, 1/4(s^2_1+s^2_2))$
