Examples of Probability (Density) Functions

There is an infinite number of possible PDFs. However, some specific forms have been very useful for describing some specific processes and have been given names. The following is a small list of examples with some discrete and several continuous distributions. The list is intended to give an overview of distributions that are often mentioned in scientific work, and some of them will be discussed again in later chapters.

Most examples are discussed as 1-dimensional distributions, except the last example, which is a higher-dimensional distribution. Again, we need to keep in mind that machine learning is mostly concerned with high-dimensional cases, so these distributions only act as a starting point for illustrating some ideas.

Bernoulli distribution

A Bernoulli random variable is a variable from an experiment that has two possible outcomes:

  • A success with probability pp
  • A failure with probability (1 − pp)

Probability function:
     P(success)=p \space \space \space \space \space P(\text {success})=p
P(failure)=1p P(\text {failure})=1-p
mean: pp
variance: p(1p)p(1-p)

Multinomial distribution

This is the distribution of outcomes in nn trials that have kk possible outcomes. The probability of each outcome is therefore pip_i.

Probability function:

P(xi)=n!i=1k(pixi/xi!)P\left(x_{i}\right)=n ! \prod_{i=1}^{k}\left(p_{i}^{x_{i}} / x_{i} !\right)

mean: npin p_{i}
variance: npi(1pi)n p_{i}\left(1-p_{i}\right)

Binomial distribution

An important example is a binomial distribution (k=2)(k = 2), which describes the number of successes in nn Bernoulli trials with probability of success pp. Note that the binomial coefficient is defined as follows:

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