# ExpectedValue

As of Version 8.0, ExpectedValue has been superseded by Expectation and NExpectation.

ExpectedValue[f,list]

gives the expected value of the pure function f with respect to the values in list.

ExpectedValue[f,list,x]

gives the expected value of the function f of x with respect to the values of list.

ExpectedValue[f,dist]

gives the expected value of the pure function f with respect to the symbolic distribution dist.

ExpectedValue[f,dist,x]

gives the expected value of the function f of x with respect to the symbolic distribution dist.

# Details and Options • For the list , the expected value of f is given by .
• For a continuous distribution dist, the expected value of f 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 expected value of f is given by where is the probability mass function of dist and summation is over the domain of dist.
• The following option can be given:
•  Assumptions \$Assumptions assumptions to make about parameters

# Examples

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## Basic Examples(3)

Find the expected value of in a Poisson distribution:

Use a pure function:

Expected value for a list:

## Scope(3)

Compute the expected value of any function:

Do the computation numerically:

Obtain expectations with conditions:

## Options(1)

### Assumptions(1)

Obtain results correct for given assumptions on symbols:

## Applications(2)

Obtain the raw moments of a distribution:

Construct a mixture density, here a Poissoninverse Gaussian mixture:

## Properties & Relations(7)

ExpectedValue of a function is the integral or sum of that function times the PDF:

ExpectedValue of for real t is the CharacteristicFunction:

ExpectedValue of a constant is the constant:

ExpectedValue of a random variable is the Mean:

ExpectedValue of the squared difference from the Mean is the Variance:

ExpectedValue for a list is a Mean:

CentralMoment is equivalent to an expected value: