# Suppose 𝑊1(𝑡) and 𝑊2(𝑡) are two independent standard Brownian motions. What is the probability that both processes are larger than 0.667 at t=1.0?

Suppose $$W_1(t)$$ and $$W_2(t)$$ are two independent standard Brownian motions. What is the probability that both processes are larger than 0.667 at t=1.0?

$$\textbf{Attempt:}$$

$$W_i(t) \sim N(0,t)$$, at time $$t=1$$, $$W_i(1) \sim N(0,1)$$; and $$\Phi(\cdot)$$ is the cdf of a standard normal random variable. We compute:

\begin{align} \mathbb{P}(W_1(1)>0.667,W_2(1)>0.667)&=\mathbb{P}(W_1(1)>0.667)\times\mathbb{P}(W_2(1)>0.667)\\ &= (\mathbb{P}(W_1(1)>0.667))^2\\ &= \Phi(-0.667)^2\\ &\approx 0.0637 \end{align}

Is that it?

• Yes, that's it!. – Kavi Rama Murthy Apr 10 at 5:38