Question: (a) Give an example of r.v.s X and Y such that FX(x) ≤ FY (x) for all x, where the inequality is strict for some x. Here FX is the CDF of X and FY is the CDF of Y. For the example you gave, sketch the CDFs of both X and Y on the same axes. Then sketch their PMFs on a second set of axes.

(b) In Part (a), you found an example of two different CDFs where the first is less than or equal to the second everywhere. Is it possible to find two different PMFs where the first is less than or equal to the second everywhere? In other words, find discrete r.v.s X and Y such that P(X = x) ≤ P(Y = x) for all x, where the inequality is strict for some x, or show that it is impossible to find such r.v.s.