Distribution of the indicator function of a Gaussian variable.

Question

Let $\xi$ be a random variable distributed according to a Normal distribution with given mean $\mu$ and standard deviation $\sigma$. Find the probability density function of

$$ \psi = c\,\mathbb{1}_{\left\{\xi\leq 0\right\}}, $$

where $\mathbb{1}_{\left\{\cdot\right\}}$ is the indicator function.

Answer 1

Since the random variable $\psi$ is discrete, its distribution has no density. Rather, one can describe it saying that $P[\psi=c]=p$ and $P[\psi=0]=1-p$, where $p=P[\xi\leqslant0]$, thus, $p=\Phi(-\mu/\sigma)$.