Probability distribution

A probability distribution is a mathematical approach to quantifying uncertainty.

There are two main classes of probability distributions: Discrete and continuous. Discrete distributions describe variables that take on discrete values only (typically the positive integers), while continuous distributions describe variables that can take on arbitrary values in a continuum (typically the real numbers).

In more advanced studies, one also comes across hybrid distributions.

Introduction to discrete distributions

Faced with a list of mutually exclusive propositions or possible outcomes, people intuitively put "degrees of belief" on the different alternatives.

For instance, consider the following 2 propositions:

• During next week there will be rain in London.
• During next week there will be no rain in London.

Based on available information about the record of past weather in England, people tend intuitively to put more "belief" in the first possibility than the second.

For another, slightly more complex example, consider the following 6 propositions:

• During next week no automobiles will enter I-45 from Galveston Island.
• During next week 1-1,000 automobiles will enter I-45 from Galveston Island.
• During next week 1,001-10,000 automobiles will enter I-45 from Galveston Island.
• During next week 10,001-100,000 automobiles will enter I-45 from Galveston Island.
• During next week 100,001-1,000,000 automobiles will enter I-45 from Galveston Island.
• During next week more than 1,000,000 automobiles will enter I-45 from Galveston Island.

Based on local information about past traffic patterns, people will intuitively distribute a "degree of belief" among the propositions.

If every "degree of belief" is a real numbers ranging from 0 to 1 and their sum is exactly 1, we have a discrete probability distribution, and each "degree of belief" is called a probability.

A discrete probability distribution is thus nothing more than a mathematically precise version of a common intuitive phenomenon, reflecting the human mind's ability to deduce and infer the physical propensities of external systems.

As a simple illustration as to how the individual probabilities may be obtained in practice, consider the expected results for a coin toss experiment. While we don't know the results for any individual toss of the coin, we can expect the results to average out to be heads half the time and tails half the time (assuming a fair coin).

Formal definition

Given a countable set S={s0, ... ,sn, ... } of mutually exclusive propositions (or possible outcomes of an experiment). Let A=[0,1}, a proper subset of the real numbers R. A discrete probability distribution is then a subset T={(s0,t0),...,(sn,tn), ...} of the cartesian product $S\times A$, such that all the ti sum to exactly 1.

Important examples

Bernoulli distribution - Each experiment is either a 1 ("success") with probability p or a 0 ("failure") with probability 1-p. An example would be tossing a coin. If the coin is fair, your probability for "success" will be exactly 50%.

An experiment where the outcome follows the Bernoulli distribution is called a Bernoulli trial.

Binomial distribution - Each experiment consists of a series of identical Bernoulli trials, f.i. tossing a coin n times, and counting the number of successes.

Uniform distribution - Each experiment has a certain finite number of possible outcomes, each with the same probability. Throwing a fair die, f.i., has six possible outcomes, each with the same probability. The Bernoulli distribution with p=0.5 is another example.

Poisson distribution - Given an experiment where we have to wait for an event to happen, and the expected remainding waiting time is independent of how long we've already waited. Then the number of events per unit time will be a Poisson distributed variable.

Geometric distribution -

Negative Binomial distribution -

Introduction to continuous distributions

We may have a set of mutually exclusive propositions or possible outcomes that is neither discrete nor finite.

For instance, consider the following infinite set of propositions:

The height of the next individual we'll meet is 0.0 meters. ... The height of the next individual we'll meet is 1.0 meters. ...

The height of an individual is a real number that in principle may take on any value.

The difficulty here is that there are too many propositions, we would never expect any one picked in advance to turn out to be the right one.

The solution is essentially a trick: One accepts that the probability for any one proposition is essentially zero (or infinitesimal), and instead concentrates on partitions of the set of propositions, i.e. intervals. The probability assigned to an interval around f.i. 1.7 meters should thus be much larger than the one assigned to a similar size interval around f.i. 2.5 meters.

Formal definition

A continuous probability distribution is a function that, whren integrated over a set representing an event, gives the probability of that event. In other words,

$p(A)=\int _{A}f\,d\mu$.

In practice, the "event" A corresponds to a range of values, say $a\leq x\leq b$, so the above integral will tae the more familiar form

$p(a\leq x\leq b)=\int _{a}^{b}f\,dx$

Important examples

Gaussian distribution - Also known as the normal distribution.

Uniform continuous distribution -

Exponential distribution - Given a sequence of events, and the waiting time between two consequitive events is independent of how long we've already waited, the time between events follows the exponential distribution.

Gamma distribution -

Rayleigh distribution -

Cauchy distribution -

Laplacian distribution -