# AsymptoticExpectation AsymptoticExpectation[expr,xdist,aa0]

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

AsymptoticExpectation[expr,{x1,x2,}dist,aa0]

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

AsymptoticExpectation[expr,vars,{a,a0,n}]

computes the asymptotic expectation to order n.

# Details and Options • Asymptotic approximations for expectations are used to estimate the values of the mean and other quantities in statistics. Examples of such uses include the law of large numbers and the study of distributions that depend on one or more parameters.
• • AsymptoticExpectation[expr,vars,aa0] computes the leading term in an asymptotic expansion for the expectation of expr. 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 a leading asymptotic approximation for the expectation of an expression:

Obtain a higher-order approximation:

Compute an asymptotic approximation for the moment-generating function of a distribution:

Obtain the first three moments of the distribution:

## Scope(9)

Compute an asymptotic expectation for a univariate continuous distribution:

Univariate discrete distribution:

Multivariate continuous distribution:

Multivariate discrete distribution:

Compute an asymptotic expectation for a TransformedDistribution:

Component mixture of normal distributions:

Parameter mixture distribution:

Marginal distribution:

Formula distribution:

## Applications(5)

Compute an asymptotic approximation for an expectation:

Compare the result with a numerical approximation:

Compute an asymptotic approximation for the mean of a distribution:

Compute an asymptotic approximation for the variance of a distribution:

Compute an asymptotic approximation for the moment-generating function of a distribution:

Obtain the first three moments of the distribution:

Compute an asymptotic approximation for the moment-generating function of a binomial distribution:

Compute an approximation for the moment-generating function of the corresponding normal distribution:

Compare the approximations for a large value of n:

## Properties & Relations(4)

Compute the asymptotic expectation of an expression in a continuous distribution:

Obtain the same result using AsymptoticIntegrate:

Compute the asymptotic expectation of an expression in a discrete distribution:

Obtain the same result using AsymptoticSum:

Use NExpectation to find the numerical value of an expectation:

Use Expectation to find the exact value of an expectation:

Obtain the asymptotic approximation using Asymptotic: