How to compare two Gaussian distributions

If I have two Gaussian distribution with the same $$\sigma$$ but different $$\mu_x$$ and $$\mu_y$$. How to calculate the $$P(x>y\ |\ \mu_x\leq\mu_y)$$? I think it's a type 1 error when I set the event "$$\mu_x>\mu_y$$" as null hypothesis?

• Another way to interperate is:what is the minimum difference between $\mu_x$ and $\mu_y$ so that we can conclude they are significantly different? So should I calculate the distribution $z = x-y$ to see if the confidence intervals of z exlude 0? Apr 6, 2020 at 19:07

Since $$\mu_x,\mu_y$$ are constants (just like $$\sigma$$. If they're not, provide their distributions), the event in the given side has no additional information. It's like $$1\leq 2$$. So, $$P(X>Y|\mu_x\leq \mu_y)=P(X>Y)$$.
$$T=X-Y$$ will be another Gaussian RV with mean $$\mu_x-\mu_y$$ and variance $$2\sigma^2$$ if assumed independent. If $$X$$ and $$Y$$ are dependent, you need to provide the joint distribution. Then,
$$P(X-Y>0)=P(T>0)=P\left(Z>\frac{0-(\mu_x-\mu_y)}{\sqrt2\sigma}\right)=\Phi\left(\frac{\mu_x-\mu_y}{\sqrt 2\sigma}\right)$$
Where $$\Phi(x)$$ is the CDF of standard Gaussian RV. See that if $$\mu_x\leq\mu_y$$, the expression inside $$\Phi$$ will be negative and the probability will be smaller than $$0.5$$, while if $$\mu_x\geq \mu_y$$, it'll be positive and the probability will be larger than $$0.5$$.