# Càdlàg property of cumulative distribution functions

I am new to stats and have struggled to understand what is the "right continuous with left limits" property of cumulative distribution functions about (i.e. càdlàg). I did my research, but still have no intuition on why that is. Could anyone please shed some light on why are the cdfs càdlàg?

• Can you tell us what you do not understand here: en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g – kjetil b halvorsen Dec 24 '16 at 13:09
• I don't get why are cdfs càdlàg as opposed to càglàd (that is the opposite). – John Halt Dec 24 '16 at 13:23
• Thats because we define them by $f(x)=P(X\le x)$. We could have defined by $F(x) =P(X < x)$ then it would be the opposite! – kjetil b halvorsen Dec 24 '16 at 13:45

The cadlag property arises naturally as a necessary property of distribution functions when defined as $F(x) = \text{Prob}(X\leq x)$), and is best understood for the case of discrete random variables.
Assume that a random variable takes the values $X \in \{1,2,3\}$. It is intuitively evident, that if we want to graph the distribution function as a line, in the interval $[1,2)$ the graph will be a straight line: probability mass arises in relation to $1$, but for all values in $(1,2)$ no additional probability mass exists. So it is natural that this graph segment includes the left end. But exactly at $2$ it should jump (since additional probability mass arises at this point exactly but not for any $2-\epsilon$ point) and then again it should remain a straight line in $[2,3)$. So we have to have something like the upper graph in the following image: Regarding the "lag" part of the property, it comes from the fact that each graph segment has to be a straight line, since in say, $(1,2)$ accumulated probability does not change as already said (always talking for a discrete random variable). This guarantees that the left limit exists.