0
$\begingroup$

A distribution function is by definition a right continuous function. For a continuous random variable however, this distinction is not really necessary because the distribution function of the continuous random variable is itself continuous and thus for all points $[0,1]$ the left limit equals the right limit.

My question is then as follows... For a continuous random variable $X$, is the following true?

$$ P[X<x] ?= P[X \leq x] $$

My reasoning is that the above equation is true because for a continuous random variable, $P[X=x] =0$.

Thanks!

$\endgroup$
4
$\begingroup$

Yes, for a continuous random variable the equality $P(X\leq x)=P(X<x)$ is true exactly for the reason you mentioned: $P(X=x)=0$.

It is useful to keep in mind that if the cumulative distribution functions is defined with the non-strict inequality (which is how it is down in most modern textbooks), then the CDF is right-continuous. However, you could also define the CDF with the strict inequality in which case it would have been left-continuous.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.