# AsymptoticProbability

AsymptoticProbability[pred,xdist,aa0]

computes an asymptotic approximation for the probability of pred centered at a0, under the assumption that x follows the probability distribution dist.

AsymptoticProbability[pred,{x1,x2,}dist,aa0]

computes an asymptotic approximation for the probability of pred centered at a0, under the assumption that {x1,x2,} follows the multivariate distribution dist.

AsymptoticProbability[pred,vars,{a,a0,n}]

computes the asymptotic probability to order n.

# Details and Options

• Asymptotic approximations for probabilities are used to study the limiting behavior of probability distributions in statistics. Examples of such uses include the central limit theorem and the approximation of the binomial distribution by a normal distribution.
• AsymptoticProbability[pred,vars,aa0] computes the leading term in an asymptotic expansion for the probability of pred. Use SeriesTermGoal to specify more terms.
• The center a0 can be any finite or infinite real or complex number.
• The order n must be a positive integer and specifies the order of approximation for the asymptotic solution. It is not related to polynomial degree.
• The following options can be given:
•  Assumptions \$Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate answers that involve conditions on parameters Method Automatic method to use PerformanceGoal \$PerformanceGoal aspects of performance to optimize SeriesTermGoal Automatic number of terms in the approximation

# Examples

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## Basic Examples(2)

Compute an asymptotic approximation for the probability of an event:

Compute a higher-order approximation:

Compute an asymptotic probability for an event in a multivariate distribution:

## Scope(8)

Compute the probability of an event in a continuous univariate distribution:

Discrete univariate distribution:

Continuous multivariate distribution:

Discrete multivariate distribution:

Component mixture of normal distributions:

Marginal distribution:

Formula distribution:

## Applications(2)

Compute an asymptotic approximation for a probability:

Compare the result with a numerical approximation:

The CDF of a distribution approaches 1 for large values of x:

## Properties & Relations(4)

Compute an asymptotic probability for an event in a continuous distribution:

Obtain the same result using AsymptoticIntegrate:

Compute an asymptotic probability for an event in a discrete distribution:

Obtain the same result using AsymptoticSum:

Use NProbability to find the numerical value of a probability:

Use Probability to find the exact value of a probability:

Obtain an asymptotic approximation using Asymptotic:

Wolfram Research (2020), AsymptoticProbability, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticProbability.html.

#### Text

Wolfram Research (2020), AsymptoticProbability, Wolfram Language function, https://reference.wolfram.com/language/ref/AsymptoticProbability.html.

#### CMS

Wolfram Language. 2020. "AsymptoticProbability." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AsymptoticProbability.html.

#### APA

Wolfram Language. (2020). AsymptoticProbability. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AsymptoticProbability.html

#### BibTeX

@misc{reference.wolfram_2024_asymptoticprobability, author="Wolfram Research", title="{AsymptoticProbability}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/AsymptoticProbability.html}", note=[Accessed: 06-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_asymptoticprobability, organization={Wolfram Research}, title={AsymptoticProbability}, year={2020}, url={https://reference.wolfram.com/language/ref/AsymptoticProbability.html}, note=[Accessed: 06-August-2024 ]}