I'm guessing you intended to say has mean and standard deviation , and has means and standard deviation .
If , then has mean
and variance
Given that and , we have
Then
and are independent, so , and
so that the variance, and hence standard deviation, are
# # #
Alternatively, if you've already learned about the variance of linear combinations of random variables, that is
then the variance of is simply the sum of the variances of and , , and so the standard deviation is again .