On average, can a person with a PhD in mathematics answer most of the questions on math.stackexchange?

My intuition said "likely not", but I decided to check. I looked at the list of top questions as of right now, 1/1/2015, 9:52am PST. I marked "No" questions that don't seem to me to be necessarily answerable by any holder of a math Ph.D.

1. What is a function? Yes. Answerable by any person with a Ph.D.
2. Points on projective line and valuations correspondence. Maybe, with effort. If one's Ph.D. is far removed from algebraic number theory, it will take some effort - reminding yourself of the definitions and spending time understanding what exactly is going on. The question happens to be not difficult (the OP managed to answer it for themselves) but most math Ph.Ds will not feel compelled to dedicate the necessary effort here, and if they do, they will have to work on it.
   
3. Does \sum_{n\ge0} \sin (\pi \sqrt{n^2+n+1}) converge/diverge? Yes. It's an exercise that any math Ph.D. should be able to do, but it's not completely trivial.
4. Confusing notation; vector spaces Yes. The question is about an exercise about a certain linear operator on the vector space of polynomials of degree at most 4, and is quite elementary. The OP simply didn't understand the notation.
   
5. Where do the coefficients belong to? Yes. Elementary facts about polynomials.
   
6. Measure theory - lebesgue's density theorem and changing order of limit Hmmmm. Not sure. The solution to the question as stated was a completely straightforward counterexample, but just understanding what's going on here requires effort. It should not be beyond the abilities of someone with a Ph.D. if they decide to tackle it, but I don't think most would, and if they do they may take a while to disentangle this.
7. Does Brouwer's theorem holds in every closed path-connected subset of \mathbb{R}^2. I'm going to go with Yes, though the question is very unclear (it mentions a set "crushed" from a ball, whatever that means). If you interpret the question reasonably as asking about sets homeomorphic to a ball, then the answer is pretty clear.
   
8. Clarification of an example. No. I mean, maybe yes, but I couldn't decipher what's going on there.
   
9. Symmetries of Archimedean Solids. No, at least not without consulting some textbooks or online summaries. Classifying finite subgroups of O_3(\mathbb{R}) is not easy without guidance.
   
10. Identify and sketch the locus of \mbox{Re} \left(\frac{z + i}{z - i} \right) = 1. Yes. Simple exercise.
    
11. Showing that the function C(b) is a compact set for |b| > 1 No, not necessarily. It's quite tricky for anyone who hasn't looked at general topology for a while.
    
12. Combinatorial definition of Hall–Littlewood polynomials (sum over SSYT?) No. The question is unclear but under any interpretation it may not be accessible to folks without good familiarity with symmetric functions and invariant theory.

I don't think 12 questions is a large enough sample, but I'm tired. Under a conservative interpretation of the tricky ones, it looks like half the problems are an easy Yes and the rest are "maybe, possibly not". Based on my experience with MSE this seems fairly typical. If you have a Ph.D. you should be able to answer about half the questions IF you're determined to dive in and help and IF the question is clear enough, which isn't always the case.