My textbook asks me the following question as a concept check:
Do you think that there are random variables which are neither discrete nor continuous? If yes, try to
construct a simple example. If no, then discuss why not.
I wrote No, how else can outcomes be measured if they aren't finite or countably infinite (which falls under discrete random variables) or if they can't be measured using a range of values (which falls under continuous random variables)? So the answer is no.
My question is, am I right? Can a random variable be described as something other then discrete or continuous? I don't think so based off my reasoning above, but I could be wrong and I'd appreciate it if someone could tell me whether I have this concept check right.
Best Answer
Of course it can, by a simple coin toss.
For example, let $X$ be a discrete distribution and $Y$ be a continuous distribution. Define the random variable $Z$ to be as follows : let $H$ be an independent (of $X$ and $Y$) Bernoulli-$\frac 12$ random variable, and define $Z = X$ if $H=0$ and $Z= Y$ if $H=1$.
Clearly $Z$ is not discrete or continuous (can be easily proven, since there are points which alone have non-zero probability by themselves, but not every point has this property as well), you may describe it as half of each. That gives rise to what is called as a mixed distribution.
The sentence "No, how can outcomes be measured ... variables)?" in your second paragraph misses the fact that an outcome , or event, need not just be a range or a set of points, but could be a mixture of both (for example, the speed of my car is either $10, 20$ or between $30$ and $40$) and hence be measured by breaking into these parts.
If one thinks of continuous random variables as those having an associated "density function" (PDF/CDF) as is usually the case, then funnily enough there is an entire class of distributions which aren't discrete, continuous, or even a mix of these. These are the singular distributions, the most well-known example being the Cantor distribution. This is not discrete (in terms of points having weights) or continuous (in terms of having a PDF/CDF) but is still a probability distribution. The buck stops here, though : every probability distribution is a mixture of a singular, continuous, and a discrete distribution.