0
$\begingroup$

It seems like different books define continuous random variables differently.

  1. pdf definition: The random variable X is continuous if a nonnegative function f exists, that is defined for all $x \in (-\infty, \infty)$ such that $P(X \in B) = \int_B f(x) dx$ (Sheldon Ross)

  2. cdf definition: The random variable X is continuous if its cdf is continuous and differentiable. Moreover, the derivative of the cdf is continuous except for a finite number of points. (Wackerly)

  3. Another cdf defintion: The random variable X is continuous if the cdf is a continuous function. (Wikipedia https://en.wikipedia.org/wiki/Random_variable#Continuous_random_variable)

  4. uncountable range: The random variable X is continuous if its range is uncountable infinite/set of possible values is uncountable infinite.

  5. uncountable sample space: The random variable X is continuous if the sample space is uncountable infinite.

My question are:

a) How do these definitions relate to each other? Are they equivalent? Why?

b) What is the difference between a continuous and an absolute continuous random variable?

c) Is it possible that the difference between continuous and discrete random variables vanishes from the point of view of measure theory? I have looked into several books about probability based on a rigid measure theoretic approach and none of them even mention the fact that there are different kinds of random variables.

d) In particular, is 4.) and 5.) equivalent? To me this doesnt seem to be necessarily the case...

e) Which is the "correct" definition from the above for a continuous random variable?

I would be happy also about answers from the point of view of measure theory, which probably answers this questions the most precisely.

$\endgroup$
  • 1
    $\begingroup$ There are too many questions here for our format, but here are some tips. (4) and (5) are woefully incorrect, although one does see statements like them in overly-simplified accounts. (1) - (3) are all forms of the Fundamental Theorem of Calculus. Thus, (a), (c), and (e) are a matter of consulting a Calculus text; (b) is answered at stats.stackexchange.com/questions/298293; and the answer to (d) is "no, but the question is of no relevance because it concerns invalid statements." $\endgroup$ – whuber Mar 24 at 18:55
  • $\begingroup$ If you would like to edit your post to focus on one of these issues, it would be interesting to answer. For instance, examples of why (4) and (5) are incorrect can be instructive. $\endgroup$ – whuber Mar 24 at 19:27
  • $\begingroup$ Thanks for your answer. I have posted a new question regarding (4) and (5). I wanted to ask, however, w.r.t. 1), 2),3): Why is 3.) a form of the Fundamental Theorem of Calculus? It doesnt say anything that the cdf would be differentiable, no? Also I have looked at your link about absolute continuous. Are 1.) and 2.) equivalent to the RV X being absolute continuous? If I understood the definition of absolute continuity correctly it says that a density exists, which is what 1.) and 2.) say, right? $\endgroup$ – guest1 Mar 25 at 10:43
  • $\begingroup$ Yes. (3) is related to an advanced version of the FTC that allows the "density" to be a measure. Measure theory provides an appropriate generalization of these concepts which exposes the fundamental ideas. Absolute continuity, in particular, is a relationship between two measures. In the usual setting the reference measure is Lebesgue measure and another (probability) measure $\mathbb P$ is absolutely continuous with respect to that provided every set to which $\mathbb P$ assigns nonzero probability has nonzero Lebesgue measure. $\endgroup$ – whuber Mar 25 at 14:24