pmf from cdf at fractional points

Question

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!

Answer 1

Guide:

• Yes, you need to do that at the place where the function is discontinuous. The probability would be the magnitude of the jump.

• Being integer is just a coincidence. It is possible for a pmf to not take positive value at any integer value at all.

• 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 $1$.