gives the numerical expectation of expr under the assumption that x follows the probability distribution dist.


gives the numerical expectation of expr under the assumption that {x1,x2,} follows the multivariate distribution dist.


gives the numerical expectation of expr under the assumption that x1, x2, are independent and follow the distributions dist1, dist2, .


gives the numerical conditional expectation of expr given pred.

Details and Options

  • xdist can be entered as x dist dist or x \[Distributed]dist.
  • exprpred can be entered as expr cond pred or expr \[Conditioned]pred .
  • NExpectation works like Expectation, except numerical summation and integration methods are used.
  • For a continuous distribution dist, the expectation of expr is given by where is the probability density function of dist and the integral is taken over the domain of dist.
  • For a discrete distribution dist, the probability of expr is given by where is the probability density function of dist and the summation is taken over the domain of dist.
  • NExpectation[expr,{x1dist1,x2dist2}] corresponds to NExpectation[NExpectation[expr,x2dist2],x1dist1] so that the last variable is summed or integrated first.
  • N[Expectation[]] calls NExpectation for expectations that cannot be done symbolically.
  • The following options can be given:
  • AccuracyGoaldigits of absolute accuracy sought
    PrecisionGoalAutomaticdigits of precision sought
    WorkingPrecisionMachinePrecisionthe precision used in internal computations
    MethodAutomaticwhat method to use
    TargetUnitsAutomaticunits to display in the output


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Basic Examples  (3)

Compute the expectation of a polynomial expression:

Compute the expectation of an arbitrary expression:

Compute a conditional expectation:

Scope  (28)

Basic Uses  (9)

Compute the expectation for an expression in a continuous univariate distribution:

Discrete univariate distribution:

Continuous multivariate distribution:

Discrete multivariate distribution:

Compute the expectation using independently distributed random variables:

Find the conditional expectation with general nonzero probability conditioning:

Discrete univariate distribution:

Multivariate continuous distribution:

Multivariate discrete distribution:

Compute the conditional expectation with a zero-probability conditioning event:

Apply N[Expectation[]] to invoke NExpectation if symbolic evaluation fails:

Find the expectation of a rational function:

Transcendental function:

Piecewise function:

Complex function:

Obtain results with different precisions:

Compute an expectation for a time slice of a Poisson process:

Find the expectation of an expression when the distribution is specified by a list:

Quantity Uses  (4)

Find expectation of quantity expressions:

Find expectations specified using QuantityDistribution:

Find conditional expectations:

Calculate expectation with QuantityMagnitude:

Equivalent calculation:

Parametric Distributions  (4)

Compute expectations for univariate continuous distributions:

Compute expectations for univariate discrete distributions:

Expectations for multivariate continuous distributions:

Expectations for multivariate discrete distributions:

Nonparametric Distributions  (2)

Using a univariate HistogramDistribution:

A multivariate histogram distribution:

Using a univariate KernelMixtureDistribution:

Derived Distributions  (9)

Compute the expectation using a TransformedDistribution:

An equivalent way of formulating the same expectation:

Find the expectation using a ProductDistribution:

An equivalent formulation for the same expectation:

Using a component mixture of normal distributions:

Parameter mixture of exponential distributions:

Truncated Dirichlet distribution:

Censored triangular distribution:

Marginal distribution:

An equivalent formulation for the same expectation:

Copula distribution:

Formula distribution:

Options  (7)

AccuracyGoal  (1)

Obtain a result with the default setting for accuracy:

Use AccuracyGoal to obtain the result with a different accuracy:

Method  (3)

Use the Method option to increase the number of recursive bisections for numerical integration:

Compare with the exact result from Expectation:

Calculate the expectation for an expression:

Obtain an estimate based on simulation:

Specify sample size:

Calculate the expectation of an expression:

This example uses NIntegrate:

Use Activate to evaluate the result:

PrecisionGoal  (1)

Obtain a result with the default setting for precision:

Use PrecisionGoal to obtain the result with a different precision:

WorkingPrecision  (1)

By default, NExpectation uses machine precision:

Use WorkingPrecision to obtain results with higher precision:

TargetUnits  (1)

Create a distribution object with quantity:

Expectation uses the quantity provided in the distribution as default:

Specify the target unit to "Hours":

Applications  (17)

Distribution Properties  (3)

Obtain a raw moment for a continuous distribution:

Obtain the mean of a discrete distribution:

Obtain the variance of a truncated distribution:

Actuarial Science  (4)

An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder's loss, , follows a distribution with density function for and 0 otherwise. Find the expected value of the benefit paid under the insurance policy:

An insurance company's monthly claims are modeled by a continuous, positive random variable , whose probability density function is proportional to where . Determine the company's expected monthly claims:

Claim amounts for wind damage to insured homes are independent random variables with common density function for and 0 otherwise, where is the amount of a claim in thousands. Suppose 3 such claims will be made. Compute the expected value of the largest of the three claims:

Let represent the age of an insured automobile involved in an accident. Let represent the length of time the owner has insured the automobile at the time of the accident. and have joint probability density function for and , and 0 otherwise. Calculate the expected age of an insured automobile involved in an accident:

Sports  (2)

A baseball player is a 0.300 hitter. Find the expected number of hits if the player comes to bat 3 times:

A basketball player shoots free throws until he hits 4 of them. His probability of scoring in any one of them is 0.7. Find the number of shots the player is expected to shoot:

Random Experiments  (2)

Four six-sided dice are rolled. Find the expectation of the minimum value:

Find the expectation of the maximum value:

Find the expectation of the sum of the three largest values. Using the identity and linearity of Expectation, you get:

A random sample of size 10 from a continuous distribution is sorted in ascending order. A new random variate is generated. Find the probability that the 11^(th) sample falls between the fourth and fifth smallest values in the sorted list:

The probability equals and is independent of :

It is also independent of the distribution:

Risk Analysis  (2)

Study the tail value at risk (TVaR) for the exponential distribution:

Value at risk may underestimate possible losses. Consider two models for stock log-returns:

Fix parameter so that values at risk at the 99.5% level are equal:

Now compute the expected losses in both models, given that they exceed the value at risk:

The losses are actually bigger in the second model:

Other Applications  (4)

A drug has proven to be effective in 40% of cases. Find the expected number of successes when applied to 100 cases:

Assuming stock logarithmic return follows a stable distribution, find the value at risk at the 95% level:

Compute the 95% value at risk point loss of the current S&P 500 index value, assuming the above distribution:

Find the expected shortfall of logarithmic return:

Compute the associated point loss:

A site has mean wind speed 7 m/s and Weibull distribution with shape parameter 2:

The resulting wind speed distribution over a whole year:

The power curve for a GE 1.5 MW wind turbine:

The total mean energy produced over the course of a year is then 4.3 GWh:

Estimate the distribution of the lengths of human chromosomes:

The expected chromosome length, given that the length is greater than the mean:

Properties & Relations  (7)

The expectation of an expression in a continuous distribution is defined by an integral:

The expectation of an expression in a discrete distribution is defined by a sum:

Mean, Moment, Variance, and other properties are defined as expectations:

Use Expectation to find a symbolic expression for an expectation:

N[Expectation[]] is equivalent to NExpectation if symbolic evaluation fails:

Use AsymptoticExpectation to find an asymptotic approximation of an expectation:

Compute the probability of an event:

Obtain the same result using NExpectation:

Possible Issues  (1)

NExpectation may fail without a warning message in the presence of symbolic parameters:

Expectation gives a closed-form result for this example:

Wolfram Research (2010), NExpectation, Wolfram Language function,


Wolfram Research (2010), NExpectation, Wolfram Language function,


@misc{reference.wolfram_2020_nexpectation, author="Wolfram Research", title="{NExpectation}", year="2010", howpublished="\url{}", note=[Accessed: 28-February-2021 ]}


@online{reference.wolfram_2020_nexpectation, organization={Wolfram Research}, title={NExpectation}, year={2010}, url={}, note=[Accessed: 28-February-2021 ]}


Wolfram Language. 2010. "NExpectation." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2010). NExpectation. Wolfram Language & System Documentation Center. Retrieved from