 Uniform distribution with an atom at zero

For $0\le q\le 1$ and $r>0$, this mixed probability distribution has probability mass $1-q$ at zero and ordinary probability density function $q/r$ on $(0,r)$. That ordinary density integrates to $q$, not to one; conditioned on $S>0$, the law is the uniform distribution on $(0,r)$. Its cumulative distribution function is zero for $s<0$, $1-q+qs/r$ for $0\le s<r$, and one for $s\ge r$.

For each positive integer $k$, integration gives $\mathbb{E}[S^k]=q{r}^k/(k+1)$. Thus the expected value is $qr/2$ and the variance is $q{r}^2/3-q^2{r}^2/4$. The nonzero atom contributes to normalization even though it contributes zero to positive moments. At $q=0$, conditioning on positive delay is undefined because the conditioning event has probability zero.