I'm having trouble with this question since mean is unknown. If the mean is centered around 0, I think I can solve.
Let be with ∼(, 1). Define a set of new random variables X such that
X={1 if >0, 0 if <=0
Question is: Derive the distribution of X and show that it depends on .
$X_i$ takes only two values $0$ and $1$. So finding the distribution of $X_i$ is just finding the probabilities that this variable takes the values $0$ and $1$.
$P(X_i=1)=P(Z_i >0)=\frac 1 {\sqrt {2\pi}}\int_0^{\infty} e^{-(x-\mu)^{2}/2} dx =\frac 1 {\sqrt {2\pi}}\int_{-\mu}^{\infty} e^{-y^{2}/2} dy=1-\Phi (-\mu)$ and $P(X_i=0)=1-P(X_i=1)=\Phi (-\mu)$ where $\Phi$ is the standard normal distribution function.