Mixed distribution problem where $F(x) = aF_d(x) + (1-a)F_c(x)$ Let $F(x) = aF_d(x) + (1-a)F_c(x)$ be the CDF of a mixed type, where

$F(x) = $ $\begin{cases} 0,    \; \;x<0,\\ {(x+1)\over 8} \;\    0\le x < 5, \\ 1, \ x\ge 5, \end{cases}$

$F_d(x)$ and $F_c(x)$ are the CDFs of discrete and continuous type, respectively.
a) Give either the discrete CDF $F_d(x)$ or the discrete pdf $f_d(x).$
b) Calculate $E(X)$, where $X$ has the $F(x)$ given above.
I absolutely do not get mixed distributions.  I don't know how to calculate them.  Can anyone help me?

First thing to search for are step discontinuities in the CDF, as found in discrete distributions  Oh, there they are at $x=0$ and $x=5$. $$F(0^-)=0, F(0^+)=0+\tfrac 18\\ F(5^-)=\tfrac 68, F(5^+)=\tfrac 68+\tfrac 28$$  What does that tell you about the discrete component? What about the coefficient $a$?

$$a\,f_d(x) =\begin{cases} 1/8 &:& x=0\\ 2/8 &:& x=5\\0&:&\text{elsewhere}\end{cases}$$ 

So...

