Mixed probability distribution ($\mu =\alpha {\mu }_d+(1-\alpha ){\mu }_c$)

A mixed probability distribution combines a discrete probability distribution with a continuous probability distribution. For example, put probability $1/4$ at zero and probability $3/4$ into a uniform distribution on $(0,1)$. Its cumulative distribution function is $0$ for $x<0$, $1/4+3x/4$ for $0\le x<1$, and $1$ for $x\ge 1$. The jump at zero and the continuous increase both contribute; neither an ordinary probability density function alone nor a probability mass function alone describes the whole law.

Let $c>0$, $0\le p\le 1$ and $\lambda >0$. With probability $p$, take $Y=-c$; otherwise draw an exponential distribution of rate $\lambda$. This mixed probability distribution has a negative atom and a continuous nonnegative component. Its moments are $\mathbb{E}[Y]=-pc+(1-p)/\lambda$ and $\mathbb{E}[Y^2]=p{c}^2+2(1-p)/{\lambda }^2$. The law of total variance equivalently gives $\mathrm{Var}(Y)=(1-p)/{\lambda }^2+p(1-p)(c+1/\lambda {)}^2$.

When $p<1$, conditioning on $Y\ge 0$ selects exactly the exponential component, so $P(Y\ge t\mid Y\ge 0)=e^{-\lambda t}$ for $t\ge 0$. An ordinary probability density function for the continuous component integrates to $1-p$, not one; the missing mass is the atom, not a normalization error.

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.