# 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

## 1 Answer

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:

We see that the distribution function in this case emerges as right-continuous and it cannot be left-continuous if it is to maintain its ability to represent correctly how probability mass accumulates here. This "cad" property also guarantees the existence of the right limit so there is no point in saying that the function is "lad".

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.

• In light of a comment by @Kjetil Halvorsen, it would seem useful--practically essential--to remind the reader that this explanation derives from a particular (and somewhat arbitrary) definition of the distribution function and to provide the definition you are using. – whuber Dec 24 '16 at 17:54
• That is indeed as clear an explanation as it gets. Thanks for taking the time to explain such a basic concept in such detail! – John Halt Dec 24 '16 at 20:54