Distribution of a Brownian Increment

Brownian increments over an interval of length $h$ are normal with mean $0$ and variance $h$.

The interval here has length $5-2=3$.

1. 1. Apply the increment property

   For standard Brownian motion, $B_t-B_s\sim N(0,t-s)$ for $s<t$, independent of the past.

2. 2. Substitute the interval length

   With $s=2$ and $t=5$, the increment has variance $5-2=3$.

   $$B_5-B_2\sim N(0,3)$$

Answer: N(0,3)