Let $X$ follow a normal distribution $N(\mu,\sigma^2)$ and $\varepsilon\in\mathbb{R}$. Then define $$Y=\frac{X-\mu}{\sigma}\,\text{ and }\,Y_\varepsilon=\frac{X-(\mu+\varepsilon)}{\sigma}$$ I want to know $\operatorname{Cov}(Y,Y_\varepsilon)$. My attempt:
$$\begin{align*}\operatorname{Cov}(X,X)&=\sigma^2=\operatorname{Cov}(\sigma Y+\mu,\sigma Y_\varepsilon+\mu+\varepsilon)\\&=\sigma^2\operatorname{Cov}(Y,Y_\varepsilon)\end{align*}$$
from which I conclude that $\operatorname{Cov}(Y,Y_\varepsilon)$ must be $1$. Is that correct?
Yes it is correct.
Observe that we can write $Y_{\epsilon}=Y+c$ where $c$ denotes a constant and can be looked at as a degenerated random variable.
Then the bilinearity of $\mathsf{Cov}$ tells us that:$$\mathsf{Cov}(Y,Y_{\epsilon})=\mathsf{Cov}(Y,Y+c)=\mathsf{Cov}(Y,Y)+\mathsf{Cov}(Y,c)=\mathsf{Var}(Y)+0=1+0=1$$