With 10 integers available,
has PMF
We're interested in the statistics of the new random variable
. To do this, we need to know the PMF for
. This isn't too hard to find.
Since the PMF for
gives a value of
whenever
is an integer between 0 and 9, it follows that
must also be an integer for the PMF to give the identical value of
. This means
Now the mean (expectation) is
The variance would be
The standard deviation is the square root of the variance, so you have