Suppose you want to check whether

has a second order stochastic dominance over

. So you would like to consider the following function:

and check whether

If their CDF is nice enough,

will be differentiable and you can try to locate the local minimum points of

, and check whether they are all non-negative. By differentiation, it is easy to see that all critical point(s)

need to satisfy

and for those critical points will be either local maximum or local minimum.

In particular, if you assume that

and

,

Now we see that if their variances are equal

, their CDFs will not intersect each other (trivial case: CDFs are overlapping if their mean also equal). So you would like to check whether

as expected.

When the variances are not equal

, the above equation will give one and only one root. That means the function

have one critical point only:

So now you need to verify the following:

1.

and this ensure

. From the definition it ensure that

will be strictly increasing up to

and strictly decreasing until

. And thus we would also need to verify

2.

For the first condition it can be similarly translated as

and this is equivalent to

(as expected; a larger variance ensure a thicker tail)

For the second condition, note that

Therefore

As a result, the conditions will be

and

and this should be enough for normal distribution.

The issue becomes interesting if you have the two independent random sample and want to test the stochastic dominance. Again if you have the normality assumption, you may have the above condition as your null hypothesis.