thus for any numbers and such that , we take the angle such that and , and the linear combination of will be Normal(0,1).

As a consequence, the sum of two independent standard Normals will be Normal(0,2). (Taking

More generally suppose we want to consider the sum
of two independent
Normals that are not standardized:

We know through the chapter on expectations and variances that
the sum of these two independent random variables
will have expectation
and variance
.
Now we consider the standardized random variable

this can be rewritten as the sum :

which is of the form , with , and