test if two normally distributed random variables are equal [duplicate]

Question

Say, I have two normally distributed random variables: $X \sim \mathcal N(\mu_x, \sigma_x^2), Y \sim \mathcal N(\mu_y, \sigma_y^2)$.

I want to test if $\mu_x = \mu_y$. Further, I assume that these variables are independent of each other. Then, I believe the test statistic would be:

${\mu_x - \mu_y}\over{\sqrt{\sigma_x^2 + \sigma_y^2)}}$

EDIT:

I want to be a bit more specific. In a journal article they test if two regression coefficients from two DIFFERENT regressions are equal.

So, I want to test if two estimated OLS coefficients from DIFFERENT regression models (the sample size is identical) are different from each other. I assume that the coefficients are independent.

Answer 1

Close, your denominator could be $\sqrt{(\sigma_x^2 + \sigma_y^2)\over{2}}$ if you had equal sample sizes, you assumed that $\sigma_x^2 \approx \sigma_y^2$, and were trying to get a better estimate of the true variance for the two variables. But as @Juho Kokkala pointed out it is sort of a silly question give that you know the population $\mu$ and $\sigma$. The solution above can be expanded by replacing $\mu$ with the sample mean $M$ and $\sigma$ with the sample estimate of the population variance $s^2$.