Is there such thing as a discrete bimodal distribution, and how do I go about hypothesising a distribution for my data?

I've got discrete data (highest education level achieved to be exact, where each level has an associated integer, from 1 to 7, with a higher number corresponding to a higher education level). The two peaks I'm expecting (for my population) is at A-levels and having completed a degree. Between them is having started but not having finished a degree, which I'm not expecting a huge deal of my population to put as their highest level.

I'm after a discrete distribution, finite support (1,2,3,4,5,6,7,8), who's pmf follows the rough shape of a bimodal distribution. Does this exist? Can I just make one up?

I've never worked with real data before. When you're looking to assign a distribution to some data such as this (real life, messy, probably not entirely representative data) how do you go about fitting a distribution? Could anyone suggest reading material for this?

• Could you explain why you feel a need to fit some kind of distribution to your data? What would that accomplish? – whuber Jul 14 '17 at 19:34
• Certainly there's such a thing as a discrete bimodal distribution -- plainly you were looking at one before you generated the above description, so questioning their existence would seem pointless. However, I agree with whuber, if you actually want an explicit functional form for the pmf, then I'd have to wonder why you'd necessarily need to do that. What function would that serve? If you need it for some modelling or something, why would (say) a general multinomial not do? – Glen_b Jul 15 '17 at 0:44