How to calculate PDF given CDF? (and related problems) I know how to find the CDF given the PDF. And I know that I must differentiate for this problem above, but I'm not sure how to go about solving these problems
(a) What is a CDF? &nbsp; You should have a list of properties that identify a function as a valid cumulative distribution function. &nbsp; Which of these is affected by the value of term $a$? &nbsp; What value of $a$ satisfies that property?

(b) Yes, indeed, differentiating the CDF will give the pdf except if there is a step discontinuity. &nbsp; Check to see if that is the case, and take steps to deal with having a mixed distribution.
The CDF, since it is the integral of the PDF, is always continuous.  Here that means that $\lim_{x\to 0^+} \frac{a- e^{-x}}{2}= \lim_{x\to 0^-} \frac{1}{2}x+ \frac{1}{2}$.  What does a have to be so those two limits are the same?