Do all random variables' probability distributions have entropy? Entropy of probability distributions is the weighted average of the log probabilities of each observation of a random variable. Does this mean that every random variable that has a probability distribution function (p.d.f.) can be measured with entropy? Are there types of pdf's that cannot be measured with entropy?
How about random variables that have a cumulative distribution function (c.d.f.) only? Does entropy apply somehow to those with only a cdf?
 A: If you mean differential entropy for continuous distributions then there are distributions whose entropy is undefined. So, the answer is NO.
For instance, look at the Warning on p.21 and the Footnote 4 in Polyanskiy and Wu, “Lecture notes on information theory”:
Shannon was probably the first who came up with a notion of entropy in information theory. So, he came up with a formula for discrete entropy, then by analogy wrote the one for continuous randoms which is called differential entropy. However, it turned out not to be the literal expansion from discrete to continuous. The latter is a different equation. Hence, I say that there are at least two different entropy equations for continuous variables.
The entropy as concept was introduced long before Shannon in statistical mechanics and thermodynamics.
A: Every random variable has a probability distribution. You can say a pdf is unknown, but I don't think you can say that it doesn't exist.
As long as you have a random variable, you can measure its entropy.
If pdf is unknown, you can estimate pdf. You can also estimate entropy directly.
For example, an estimator for entropy is this:

"Summing the log of the nearest neighbors distances: a simple estimator for the entropy from samples"


Update:
Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value [1]. But, there are also some random variables that doesn't have a pdf nor a pmf, for example Cantor distribution. So, you can also say that a pdf doesn't exist.
