Letâ€™s explain why the normal distribution is so important.

(This is a section in the notes here.)

Continue reading “The Law of Large Numbers and Central Limit Theorem”

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# Category: Probability

## The Law of Large Numbers and Central Limit Theorem

## Continuous Probability Distributions

## Discrete Probability Distributions

## Random Variables and Expectation

## Conditional Probability

## Counting Principles

## Probability and Set Operations

## What is Probability?

## A. Mathematical Notation

## Exponential Families

Letâ€™s explain why the normal distribution is so important.

(This is a section in the notes here.)

Continue reading “The Law of Large Numbers and Central Limit Theorem”

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.)

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.)

Often we are interested in the magnitude of an outcome as well as its probability. E.g. in a gambling game amount you win or loss is as important as the probability each outcome.

(This is a section in the notes here.)

(This is a section in the notes here.)

Conditional probabilities are probabilities where we have assumed that another event has occurred.

(This is a section in the notes here.)

**Counting in Probability.** If each outcome is equally likely, i.e. for all , then since

(where is the number of outcomes in the set ) it must be that

(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.

(This is a section in the notes here.)

I throw a coin times. I got heads.

This is the appendix in the notes here.

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”