Difference between revisions of "Probability distributions"
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| − | + | == Distribution Functions for Random Variables == | |
The distribution function provides important information about the probabilities associated with different values of a random variable. It can be used to calculate probabilities for specific events or to obtain other statistical properties of the random variable. | The distribution function provides important information about the probabilities associated with different values of a random variable. It can be used to calculate probabilities for specific events or to obtain other statistical properties of the random variable. | ||
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*''[[Relative frequency]] distribution'': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] (i.e. sample size). | *''[[Relative frequency]] distribution'': a [[frequency distribution]] where each value has been divided (normalized) by a number of outcomes in a [[Sample (statistics)|sample]] (i.e. sample size). | ||
*''[[Categorical distribution]]'': for discrete random variables with a finite set of values. | *''[[Categorical distribution]]'': for discrete random variables with a finite set of values. | ||
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=== Distribution Functions for Discrete Random Variables === | === Distribution Functions for Discrete Random Variables === | ||
Revision as of 18:49, 30 May 2023
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.[1][2] It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).[3]
Contents
Introduction
Probability is the science of uncertainty. It provides precise mathematical rules for understanding and analyzing our own ignorance. It does not tell us tomorrow’s weather or next week’s stock prices; rather, it gives us a framework for working with our limited knowledge and for making sensible decisions based on what we do and do not know. (citee)
Terminology
Sample space
In probability theory, the sample space refers to the set of all possible outcomes of a random experiment. It is denoted by the symbol Ω (capital omega).
- Let's consider an example of rolling a fair six-sided die. The sample space in this case would be {1, 2, 3, 4, 5, 6}, as these are the possible outcomes of the experiment. Each number represents the face of the die that may appear when it is rolled.
Random variable
Random variable takes values from a sample space. In contrast, probabilities describe which values and set of values are more likely to be taken out of the sample space. Random variable must be quantified, therefore, it assigns a numerical value to each possible outcome in the sample space.
- For example, if the sample space for flipping a coin is {heads, tails}, then we can assign a random variable Y such that Y = 1 when heads land and Y = 0 when tails land. However, we can assign any number for these variables. 0 and 1 are just more convenient.
- Because random variables are defined to be functions of the outcome s, and because the outcome s is assumed to be random (i.e., to take on different values with different probabilities), it follows that the value of a random variable will itself be random (as the name implies).
Specifically, if X is a random variable, then what is the probability that X will equal some particular value x? Well, X = x precisely when the outcome s is chosen such that X(s) = x.
- Exercise
- Suppose that a coin is tossed twice so that the sample space is S = {HH, HT, TH, TT}. Let X represent the number of heads that can come up. With each sample point we can associate a number for X as shown in Table 1. Thus, for example, in the case of HH (i.e., 2 heads), X = 2 while for TH (1 head), X = 1. It follows that X is a random variable.
Distribution Functions for Random Variables
The distribution function provides important information about the probabilities associated with different values of a random variable. It can be used to calculate probabilities for specific events or to obtain other statistical properties of the random variable.
- It gives the probability that the random variable takes on a value less than or equal to a given value.
The distribution function of a random variable X, denoted as F(x), is defined as:
- F(x) = P(X ≤ x)
where x is any real number, and P(X ≤ x) is the probability that the random variable X is less than or equal to x.
- Discrete probability distribution: for many random variables with finitely or countably infinitely many values.
- Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value.
- Frequency distribution: a table that displays the frequency of various outcomes Template:Em.
- Relative frequency distribution: a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size).
- Categorical distribution: for discrete random variables with a finite set of values.
Distribution Functions for Discrete Random Variables
If X takes on only a finite number of values x1, x2, . . . , xn, then the distribution function is given by
The following function: 
Distribution Functions for Discrete Continuous Variables
Related terms
- Support: set of values that can be assumed with non-zero probability by the random variable. For a random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} , it is sometimes denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_X} .
- Tail:[4] the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X > a} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X < b} or a union thereof.
- Head:[4] the region where the pmf or pdf is relatively high. Usually has the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a < X < b} .
- Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.
- Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
- Mode: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
- Quantile: the q-quantile is the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X < x) = q} .
- Variance: the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.
- Standard deviation: the square root of the variance, and hence another measure of dispersion.
- Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value (usually the median) is a mirror image of the portion to its right.
- Skewness: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.
- Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.
Cumulative distribution function
In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} with regard to a probability distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p} is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x) = P(X \leq x).}
The cumulative distribution function of any real-valued random variable has the properties:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x)} is non-decreasing;
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x)} is right-continuous;
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0 \le F(x) \le 1} ;
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to -\infty} F(x) = 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to \infty} F(x) = 1} ; and
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Pr(a < X \le b) = F(b) - F(a)} .
Conversely, any function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F:\mathbb{R}\to\mathbb{R}} that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.[5]
Any probability distribution can be decomposed as the mixture of a discrete, an absolutely continuous and a singular continuous distribution,[6] and thus any cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions.
Discrete probability distribution
A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values[7] (almost surely)[8] which means that the probability of any event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} can be expressed as a (finite or countably infinite) sum: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X\in E) = \sum_{\omega\in A \cap E} P(X = \omega),} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} is a countable set with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X \in A) = 1} . Thus the discrete random variables are exactly those with a probability mass function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p(x) = P(X=x)} . In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p(n) = \tfrac{1}{2^n}} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n = 1, 2, ...} , the sum of probabilities would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/2 + 1/4 + 1/8 + \dots = 1} .
A discrete random variable is a random variable whose probability distribution is discrete.
Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution.[3] When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.
Cumulative distribution function
A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take. Thus the cumulative distribution function has the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x) = P(X \leq x) = \sum_{\omega \leq x} p(\omega).}
The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.
Dirac delta representation
A discrete probability distribution is often represented with Dirac measures, the probability distributions of deterministic random variables. For any outcome Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega} , let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta_\omega} be the Dirac measure concentrated at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega} . Given a discrete probability distribution, there is a countable set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X \in A) = 1} and a probability mass function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} is any event, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X \in E) = \sum_{\omega \in A} p(\omega) \delta_\omega(E),} or in short, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_X = \sum_{\omega \in A} p(\omega) \delta_\omega.}
Similarly, discrete distributions can be represented with the Dirac delta function as a generalized probability density function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = \sum_{\omega \in A} p(\omega) \delta(x - \omega),} which means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X \in E) = \int_E f(x) \, dx = \sum_{\omega \in A} p(\omega) \int_E \delta(x - \omega) = \sum_{\omega \in A \cap E} p(\omega)} for any event Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E.} [9]
Indicator-function representation
For a discrete random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} , let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_0, u_1, \dots} be the values it can take with non-zero probability. Denote
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots}
These are disjoint sets, and for such sets
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.}
It follows that the probability that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} takes any value except for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_0, u_1, \dots} is zero, and thus one can write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} as
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)}
except on a set of probability zero, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1_A} is the indicator function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} . This may serve as an alternative definition of discrete random variables.
One-point distribution
A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a deterministic distribution. Expressed formally, the random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} has a one-point distribution if it has a possible outcome Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X{=}x)=1.} [10] All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1.
Absolutely continuous probability distribution
An absolutely continuous probability distribution is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral.[11] More precisely, a real random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} has an absolutely continuous probability distribution if there is a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f: \Reals \to [0, \infty]} such that for each interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [a,b] \subset \mathbb{R}} the probability of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} belonging to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [a,b]} is given by the integral of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f} over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle I} :[12][13] Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P\left(a \le X \le b \right) = \int_a^b f(x) \, dx .} This is the definition of a probability density function, so that absolutely continuous probability distributions are exactly those with a probability density function. In particular, the probability for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} to take any single value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a} (that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a \le X \le a} ) is zero, because an integral with coinciding upper and lower limits is always equal to zero. If the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [a,b]} is replaced by any measurable set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} , the according equality still holds: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P(X \in A) = \int_A f(x) \, dx .}
An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous.
There are many examples of absolutely continuous probability distributions: normal, uniform, chi-squared, and others.
Cumulative distribution function
Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution function. In this case, the cumulative distribution function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} has the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x) = P(X \leq x) = \int_{-\infty}^x f(t)\,dt} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f} is a density of the random variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X} with regard to the distribution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P} .
Note on terminology: Absolutely continuous distributions ought to be distinguished from continuous distributions, which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but the inverse is not true, there exist singular distributions, which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the Cantor distribution. Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions.[14]
For a more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure.
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ 3.0 3.1 Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ 4.0 4.1 More information and examples can be found in the articles Heavy-tailed distribution, Long-tailed distribution, fat-tailed distribution
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ see Lebesgue's decomposition theorem
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ [15]
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
- ↑ Chapter 3.2 of Template:Harvp
- ↑ Template:Cite web
- ↑ Spiegel, M. R., Schiller, J. T., & Srinivasan, A. (2001). Probability and Statistics : based on Schaum’s outline of Probability and Statistics by Murray R. Spiegel, John Schiller, and R. Alu Srinivasan. https://ci.nii.ac.jp/ncid/BA77714681
