Consider $N\sim\mathcal N(0,1)$ a standard normal random variable. Can we find a dependent random variable $D$, such that the random variable $M=N+D$ is still a standard normal distribution? I would also like to choose the variance of D that would be typically smaller than 1. $D$ would represent a kind of random noise that do not change the random distribution when added.
If $D$ is an independent distribution with average 0, the average is still $0$ but the variance of $D$ is added to that of $N$. This property is not true when the random variables are correlated. We might determine $D$ with its conditional distribution : $f(D | N)$
I have no idea how to solve this problem.