Behind conditional bivariate probabilities

Here we establish the characteristics of a bivariate normal distribution.  Note that this is not a standard bivariate normal distribution with independent X and Y variables.  The joint marginal distribution is therefore not in the form f(x)*f(y).  We instead have a more complex joint marginal distribution of the form f(x)*f(y|x), or symmetrically f(y)*f(x|y).  See this distribution formula build-up, below:

We can observe from the penultimate formula above that the distribution of Y, given X=x, is normally distributed.  The variable Y it shows, from the the exponent, has a mean and variance of:

How does one solve for the level of x needed to create a conditional mean of Y that is 2 and 3 standard deviations_y above this x level?

For Y at 2 standard deviations_y above the x level:

Similarly for Y at 3 standard deviations_y above the x level:

We will complement these results with a graphical simulation in a forthcoming blog note.