Continuous Probability Distributions

We consider distributions that have a continuous range of values. Discrete probability distributions where defined by a probability mass function. Analogously continuous probability distributions are defined by a probability density function.

(This is a section in the notes here.)

Continue reading “Continuous Probability Distributions”

Discrete Probability Distributions

There are some probability distributions that occur frequently. This is because they either have a particularly natural or simple construction. Or they arise as the limit of some simpler distribution. Here we cover

  • Bernoulli random variables
  • Binomial distribution
  • Geometric distribution
  • Poisson distribution.

(This is a section in the notes here.)

Continue reading “Discrete Probability Distributions”

Probability and Set Operations

(This is a section in the notes here.)

We want to calculate probabilities for different events. Events are sets of outcomes, and we recall that there are various ways of combining sets. The current section is a bit abstract but will become more useful for concrete calculations later.

Continue reading “Probability and Set Operations”

Exponential Families

The exponential family of distributions are a particularly tractable, yet broad, class of probability distributions. They are tractable because of a particularly nice [Fenchel] duality relationship between natural parameters and moment parameters. Moment parameters can be estimated by taking the empirical mean of sufficient statistics and the duality relationship can then recover an estimate of the distributions natural parameters.

Continue reading “Exponential Families”