What is a Brownian motion? State the reflection principle for Brownian motion.

Let $W=\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion. Let $M=\max _{0 \leqslant t \leqslant 1} W_{t}$. Prove

$$\mathbb{P}\left(M \geqslant x, W_{1} \leqslant x-y\right)=\mathbb{P}\left(M \geqslant x, W_{1} \geqslant x+y\right)$$

for all $x, y \geqslant 0$. Hence, show that the random variables $M$ and $\left|W_{1}\right|$ have the same distribution.

Find the density function of the random variable $R=W_{1} / M$.