Probability distribution: Difference between revisions

Revision as of 11:20, 15 November 2007

A gentle introduction to the concept

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

A simple example

When you wake up in the morning one of three thing may happen that day:

You will get hit by a meteor falling in from space.
You will not get hit by a meteor falling in from space, but you'll be struck by lightning.
Neither will happen.

Most people will usually intuit a small to zero belief in the first alternative (although it is possible, and is known to actually have occurred), a slightly larger belief in the second, and a rather strong belief in the third.

In mathematics, such intuitive ideas are captured, formalized and made precise by the concept of a discrete probability distribution.

A more complicated example

Rather than a simple list of propositions or outcomes like the one above, one may have a to deal with a continuum.

For example, consider the next new person you'll get to know. Given a way to measure height exactly, with infinite precision, how tall will he or she be?

This can be formulated as an uncountably infinite set of propositions, or as a ditto set of possible outcomes of a random experiment.

Let's look at three of these propositions in detail:

...

The person is exactly 1.7222... m tall.

...

The person is exactly 2.333... m tall.

...

The person is exactly 25.010101... m tall.

...

Clearly, we don't believe the person will be over 25 meters tall. But neither do we believe any of the other propositions. Why should any particular proposition turn out to be the exact correct one among an infinity of others?

But we still somehow feel that the first proposition listed is more "likely" than the second, which again is more "likely" than the third.

Also, we feel that some "ranges" are more likely than others, f.i. a height between 1.6 and 1.8 meters feels "likely", a height between 2.2 and 2.4 m seems possible but unlikely, and a height larger than that usually seems safe to exclude.

In mathematics, such intuitive ideas are captured, formalized and made precise by the concept of a continuous probability distribution.

A formal introduction

Discrete probability distributions

Let $S=\{...,s_{0},s_{1},...\}$ be a countable set. Let f be a function from S to $R$ such that

f(s) ∈ [0,1] for all s ∈ S
The sum $\sum _{i=-\infty }^{\infty }f(s_{i})$ exists and evaluates to exactly 1.

Then f is a probability mass distribution over the set S. The function F on S defined by

$F(s_{i})=\sum _{k=-\infty }^{i}f(s_{k})$

is said to be a discrete probability distribution on S.

Continuous probability distributions

Let f be a function from $\mathbb {R}$ to $\mathbb {R}$ such that

f is measurable and f(s) ≥ 0 for all s in $\mathbb {R}$
The Lebesgue integral $\int _{-\infty }^{\infty }f(s)ds$ exists and evaluates to exactly 1.

Then f is said to be a probability density on the real line. The function $F:(-\infty ,\infty )\rightarrow [0,1]$ defined as the integral:

$F(x)=\int _{-\infty }^{x}f(s)ds,$

is said to be a continuous probability distribution on the real line. This type of distribution is absolutely continuous with respect to the Lebesgue measure.

It should be emphasized that the above is a basic definition of probability distributions on the real line. In probability theory, probability distributions are actually defined much more generally in terms of sigma algebras and measures. Using these general definitions, one can even formulate probability distributions for general classes of abstract sets beyond the real numbers or Euclidean spaces.

Probability distributions in practice

Statistical methods used to choose between distributions and estimate parameters

In the first example in this article, one may look through medical records to find approximately how many people are known to suffer such mishaps per century, and from that information create a statistic to estimate the probabilities. A strict frequentist will stop there, most statisticians will allow non-statisticial information to be used to arrive at what would be considered the best available distribution to model the problem. Such information would include knowledge and intuition about peoples tendency to consult doctors after accidents, the comprehensiveness of the records and so on.

References

[1]Person actually hit by a meteorite.

External links

@@ Line 1: / Line 1: @@
+{{subpages}}
 A [[probability distribution]] is a mathematical approach to quantifying uncertainty.
+There are two main classes of probability distributions: Discrete and continuous. [[Discrete probability distribution|Discrete  distributions]] describe variables that take on discrete values only (typically the positive integers), while [[continuous probability distribution|continuous  distributions]] describe variables that can take on arbitrary values in a continuum (typically the real numbers).
-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.
+==A gentle introduction to the concept==
+Faced with a set of mutually exclusive propositions or possible outcomes, people intuitively put "degrees of belief" on the different alternatives.
-== Introduction to discrete distributions ==
+===A simple example===
+When you wake up in the morning one of three thing may happen that day:
+*You will get hit by a meteor falling in from space.
+*You will not get hit by a meteor falling in from space,  but you'll be struck by lightning.
+*Neither will happen.
-Faced with a list of mutually exclusive propositions or possible outcomes, people intuitively put "degrees of belief" on the different alternatives.
+Most people will usually intuit a small to zero belief in the first alternative (although it is possible,  and is known to actually have occurred),  a slightly larger belief in the second,  and a rather strong belief in the third.
-For instance, consider the following 2 propositions:
+In mathematics,  such intuitive ideas are captured,  formalized and made precise by the concept of a [[discrete probability distribution]].
-*During next week there will be rain in London.
+===A more complicated example===
-*During next week there will be no rain in London.
+Rather than a simple list of propositions or outcomes like the one above,  one may have a to deal with a continuum.
-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 example,  consider the next new person you'll get to know.  Given a way to measure height ''exactly, with infinite precision'', how tall will he or she be?
-For another, slightly more complex example, consider the following 6 propositions:
+This can be formulated as an [[uncountably infinite set]] of propositions,  or as a ditto set of possible outcomes of a [[random experiment]].
-*During next week no automobiles will enter I-45 from Galveston Island.
+Let's look at three of these propositions in detail:
-*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.
+...
+*The person is exactly 1.7222... m tall.
+...
+*The person is exactly 2.333... m tall.
+...
+*The person is exactly 25.010101... m tall.
+...
-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]].
+Clearly,  we don't believe the person will be over 25 meters tall.  But neither do we believe any of the other propositions.  Why should any particular proposition turn out to be the ''exact'' correct one among an [[infinity]] of others?
-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.
+But we still somehow feel that the first proposition listed is more "likely" than the second,  which again is more "likely" than the third.
-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).
+Also,  we feel that some "ranges" are more likely than others,  f.i. a height ''between'' 1.6 and 1.8 meters feels "likely",  a height ''between'' 2.2 and 2.4 m seems possible but unlikely,  and a height larger than that usually seems safe to exclude.
+In mathematics,  such intuitive ideas are captured,  formalized and made precise by the concept of a [[continuous probability distribution]].
-=== Formal definition===
-Given a countable set S={s0, ... ,sn, ... } of mutually exclusive propositions (or possible outcomes of an experiment).
+==A formal introduction==
-Let A=[0,1}, a proper subset of the [[real number]]s '''R'''.
-A discrete probability distribution is then a subset T={(s0,t0),...,(sn,tn), ...} of the cartesian product <math>S \times A</math>,  such that all the ti sum to exactly 1.
+===Discrete probability distributions===
+Let <math>S=\{ ..., s_0, s_1, ...\}</math> be a countable set.
+Let f be a function from S to <math>R</math> such that
+*f(s) &isin; [0,1] for all s &isin; S
+*The sum <math> \sum_{i=-\infty}^{\infty} f(s_i) </math> exists and evaluates to exactly 1.
-=== Important examples ===
+Then f is a '''probability mass distribution''' over the set S. The function F on S defined by
-[[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%.
+<math>
+F(s_i)=\sum_{k=-\infty}^{i} f(s_k)
+</math>
-An experiment where the outcome follows the Bernoulli distribution is called a Bernoulli trial.
+is said to be a '''discrete probability distribution''' on S.
-[[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.
+===Continuous probability distributions===
-[[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.
+Let f be a function from <math>\mathbb{R}</math> to <math>\mathbb{R}</math> such that
+*f is [[measurable function|measurable]] and f(s) ≥ 0 for all s in <math>\mathbb{R}</math>
+*The [[Lebesgue integral]] <math> \int_{-\infty}^{\infty} f(s)ds</math> exists and evaluates to exactly 1.
-[[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.
+Then f is said to be a '''probability density''' on the real line. The function <math>F:(-\infty,\infty)\rightarrow [0,1]</math> defined as the integral:
-[[Geometric distribution]] -
+<math>F(x)=\int_{-\infty}^{x}f(s)ds,</math>
-[[Negative Binomial distribution]] -
+is said to be a '''continuous probability distribution''' on the real line. This type of distribution is [[absolute continuity|absolutely continuous]] with respect to the [[Lebesgue measure]].
+It should be emphasized that the above is a basic definition of probability distributions on the real line. In [[probability theory]], probability distributions are actually defined much more generally in terms of [[sigma algebra|sigma algebras]] and [[measure|measures]]. Using these general definitions, one can even formulate probability distributions for general classes of abstract sets beyond the real numbers or Euclidean spaces.
+==Probability distributions in practice==
+===Statistical methods used to choose between distributions and estimate parameters===
+In the first example in this article, one may look through medical records to find approximately how many people are known to suffer such mishaps per century,  and from that information create a [[statistic]] to [[statistical parameter estimation|estimate]] the probabilities.  A strict [[frequentist interpretation of probability|frequentist]] will stop there, most statisticians will allow non-[[statistics|statisticial]] information to be used to arrive at what would be considered the best available distribution to model the problem.  Such information would include knowledge and intuition about peoples tendency to  consult doctors after accidents, the comprehensiveness of the records and so on.
-== Introduction to continuous distributions ==
+==References==
+*[http://www.time.com/time/magazine/article/0,9171,821063,00.html]Person actually hit by a meteorite.
-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:
+==See also==
+*[[Discrete probability distribution]]
+*[[Continuous probability distribution]]
+*[[Probability]]
+*[[Probability theory]]
+*[[Entropy of a probability distribution]]
-*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,  in practice intervals in R. 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,
-<math>p(A) = \int_A f\,d\mu</math>.
-In practice, the "event" A corresponds to a range of values, say <math>a \le x \le b</math>, so the above integral will tae the more familiar form
-<math>p(a \le x \le b) = \int_a^b f\,dx</math>
-=== 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]] -
-==References==
-==See also==
-*[[Entropy of a probability distribution]]
 ==Related topics==
@@ Line 125: / Line 106: @@
 ==External links==
-[[Category:Mathematics Workgroup]]

Probability distribution: Difference between revisions

Revision as of 11:20, 15 November 2007

Contents

A gentle introduction to the concept

A simple example

A more complicated example

A formal introduction

Discrete probability distributions

Continuous probability distributions

Probability distributions in practice

Statistical methods used to choose between distributions and estimate parameters

References

See also

Related topics

External links

Navigation menu

Probability distribution: Difference between revisions

Revision as of 11:20, 15 November 2007

A gentle introduction to the concept

A simple example

A more complicated example

A formal introduction

Discrete probability distributions

Continuous probability distributions

Probability distributions in practice

Statistical methods used to choose between distributions and estimate parameters

References

See also

Related topics

External links

Navigation menu

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