If I have $X = 5(B_t - B_s)$
Does this have a distribution of $\sim \text{N}(0,25(t-s))$ ?
Since $B_t - B_s$ has distribution $\sim \text{N}(0,t-s)$
Then $X = \mu \cdot 0 + \sigma_1 Z$ where $Z \sim \text{N}(0,1)$ and $\sigma_1 = \sqrt{t-s}$
So $5X = \sigma_2 Z$ where $\sigma_2 = 5 \cdot \sigma_1$
so $5X \sim \text{N}(0, \sigma_2^2) = \sim \text{N}(0, 25(t-s))$
Would appreciate if someone could check this result and give it a thumbs up.