Say we are given a normal distribution with mean $\mu$ and standard deviation $\sigma$, where $\mu$ is the expectation in time period $T$.
Now, say you need to use the distribution over a shorter time period, say $t, t < T$.
It stands to reason that the expectation over this time period $t$ is $\frac{t}{T}\mu$. However, what can be said about the standard deviation in this period? Will it remain $\sigma$?