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I'm wondering what are some important distributions to know the basic properties of (pmf/pdf, mean, variance) specifically as it comes to times series analysis, financial analysis, and machine learning.

I've gone through the basic ones (Bernoulli, Binomial, Geometric, Poisson, Normal), and out of curiosity I looked at this list which seems like there's way too many for a regular person to be able to know.

https://en.wikipedia.org/wiki/List_of_probability_distributions

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    I think this is rather subjective and usually it is enough to be able to check the distribution when you need to. – 2017-02-20

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Which distributions you should know depends on the topics you're investigating. For time series analysis, financial analysis and machine learning, I teach and would include:

  1. 1D and nD Gaussian
  2. Bernoulli
  3. Exponential
  4. Rayleigh
  5. Maxwell
  6. Gamma
  7. Beta
  8. Poisson
  9. Binomial and Multinomial

Once you know these, you can easily extend your knowledge to closely related distributions (e.g., LogNormal, etc.).