# pmf from cdf at fractional points

I am new to probability and am taking an introductory probability course this semester. We have a cdf as:

\begin{align}F(x) = 0 \space &\text{ if } x <1, \\ \frac14 \space &\text{ if } 1\le x <\frac53, \\ \frac13 \space &\text{ if } \frac53\le x <2, \\ \frac35 \space &\text{ if } 2\le x <\frac52, \\ 1 \space &\text{ if } x \ge\frac52.\end{align}

I calculated $$P(X=1) = F(1\le x< \frac53) - F(X<1) = 1/4$$

and $$P(X=2) = F(2\le x<5/2) - F(5/3\le x<2) = \frac{4}{15}$$

My question is do I need to find the pmfs at $$5/3(P(X=\frac53)) \space \text{and } 5/2(P(X=\frac52))$$ also?

At this link a comment by SchrodingersCat mentions we "taking the respective differences for the points where the function is discontinuous". So I am curious if we should do the same at non-integer points too?

Thanks!

• Also remember to state that for the remaining points without jump, the corresponding probability would be $$0$$.
• After you compute your pmf, do a sanity check that the sum of the probability adds up to $$1$$.