Probability theory: Difference between revisions

Revision as of 12:25, 27 February 2023

Revision as of 13:25, 27 February 2023

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The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1, that is, absolute certainty.

When doing calculations using the outcomes of an experiment, it is necessary that all those [[elementary event]]s have a number assigned to them. This is done using a [[random variable]]. A random variable is a function that assigns to each elementary event in the sample space a [[real number]]. This function is usually denoted by a capital letter.<ref>{{Cite book |title =Introduction to Probability and Mathematical Statistics |last1 =Bain |first1 =Lee J. |last2 =Engelhardt |first2 =Max |publisher =Brooks/Cole |___location =[[Belmont, California]] |page =53 |isbn =978-0-534-38020-5 |edition =2nd |date =1992 }}</ref> In the case of a dice, the assignment of a number to a certain elementary events can be done using the [[identity function]]. This does not always work. For example, when [[coin flipping|flipping a coin]] the two possible outcomes are "heads" and "tails". In this example, the random variable ''X'' could assign to the outcome "heads" the number "0" (<math>X(heads)=0</math>) and to the outcome "tails" the number "1" (<math>X(tails)=1</math>).

===Discrete probability distributions===

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{{em|Discrete probability theory}} deals with events that occur in [[countable]] sample spaces.

Examples: Throwing [[dice]], experiments with [[deck of cards|decks of cards]], [[random walk]], and tossing [[coin]]s.

{{em|Classical definition}}:

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{{em|Modern definition}}:

If the sample space of a random variable ''X'' is the set of [[real numbers]] (<math>\mathbb{R}</math>) or a subset thereof, then a function called the {{em|[[cumulative distribution function]]}} (or {{em|cdfCDF}}) <math>F\,</math> exists, defined by <math>F(x) = P(X\le x) \,</math>. That is, ''F''(''x'') returns the probability that ''X'' will be less than or equal to ''x''.

The cdfCDF necessarily satisfies the following properties.

# <math>F\,</math> is a [[Monotonic function|monotonically non-decreasing]], [[right-continuous]] function;

# <math>\lim_{x\rightarrow -\infty} F(x)=0\,;</math>

# <math>\lim_{x\rightarrow \infty} F(x)=1\,.</math>

The random variable <math>X</math> is said to have a continuous probability distribution if the corresponding cdfCDF <math>F</math> is continuous. If <math>F\,</math> is [[absolutely continuous]], i.e., its derivative exists and integrating the derivative gives us the cdfCDF back again, then the random variable ''X'' is said to have a {{em|[[probability density function]]}} or ({{em|pdfPDF}}) or simply {{em|density}} <math>f(x)=\frac{dF(x)}{dx}\,.</math>

For a set <math>E \subseteq \mathbb{R}</math>, the probability of the random variable ''X'' being in <math>E\,</math> is

:<math>P(X\in E) = \int_{x\in E} dF(x)\,.</math>

In case the probability density functionPDF exists, this can be written as

:<math>P(X\in E) = \int_{x\in E} f(x)\,dx\,.</math>

Whereas the ''pdf'' exists only for continuous random variables, the ''cdfCDF'' exists for all random variables (including discrete random variables) that take values in <math>\mathbb{R}\,.</math>

These concepts can be generalized for [[Dimension|multidimensional]] cases on <math>\mathbb{R}^n</math> and other continuous sample spaces.

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Given any set <math>\Omega\,</math> (also called {{em|sample space}}) and a [[sigma-algebra|σ-algebra]] <math>\mathcal{F}\,</math> on it, a [[measure (mathematics)|measure]] <math>P\,</math> defined on <math>\mathcal{F}\,</math> is called a {{em|probability measure}} if <math>P(\Omega)=1.\,</math>

If <math>\mathcal{F}\,</math> is the [[Borel algebra|Borel σ-algebra]] on the set of real numbers, then there is a unique probability measure on <math>\mathcal{F}\,</math> for any cdfCDF, and vice versa. The measure corresponding to a cdfCDF is said to be {{em|induced}} by the cdfCDF. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theoretic approach free of fallacies.

The ''probability'' of a set <math>E\,</math> in the σ-algebra <math>\mathcal{F}\,</math> is defined as

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In probability theory, there are several notions of convergence for [[random variable]]s. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions.

;Weak convergence: A sequence of random variables <math>X_1,X_2,\dots,\,</math> converges {{em|weakly}} to the random variable <math>X\,</math> if their respective cumulative ''distribution functions''CDF <math>F_1,F_2,\dots\,</math> converge to the cumulative distribution functionCDF <math>F\,</math> of <math>X\,</math>, wherever <math>F\,</math> is [[continuous function|continuous]]. Weak convergence is also called {{em|convergence in distribution}}.

:Most common shorthand notation: <math>\displaystyle X_n \, \xrightarrow{\mathcal D} \, X</math>