Adding 2 to each value of the random variable makes a new random variable
. Its mean would be
since expectation is linear, and the expected value of a constant is that constant. is the mean of
, so the new mean would be
The variance of a random variable is
so the variance of would be
We already know , so simplifying above, we get
Standard deviation is the square root of variance, so .
so the standard deviation remains unchanged at 3.
NB: More generally, the variance of for
is
but the variance of a constant is 0. In this case, , so we're left with
, as expected.