ExpectedValue

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`ExpectedValue`

As of Version 8.0, ExpectedValue has been superseded by Expectation and NExpectation.

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ExpectedValue[f,list]

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

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ExpectedValue[f,list,x]

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

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ExpectedValue[f,dist]

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

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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

open allclose all

Basic Examples (3)Summary of the most common use cases

Find the expected value of in a Poisson distribution:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-b8jmn0

Out[1]=1

Use a pure function:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-f9mz7p

Out[1]=1

Expected value for a list:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-cbwncz

Out[1]=1

Scope (3)Survey of the scope of standard use cases

Compute the expected value of any function:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-cnnfq

Out[1]=1

Do the computation numerically:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-hbkpj4

Out[1]=1

Obtain expectations with conditions:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-cy28hz

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-kb9rl2

Out[2]=2

Options (1)Common values & functionality for each option

Assumptions (1)

Obtain results correct for given assumptions on symbols:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-u8w8t

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-g8n09d

Out[2]=2

Applications (2)Sample problems that can be solved with this function

Obtain the raw moments of a distribution:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-gjd0w3

Out[1]=1

Construct a mixture density, here a Poisson‐inverse Gaussian mixture:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-gq71xc

Out[1]=1

Properties & Relations (7)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-nfzfd

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-lj99yt

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0rs07jqyi-g90uia

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0rs07jqyi-feeiza

Out[4]=4

ExpectedValue of for real t is the CharacteristicFunction:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-lzo4k4

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-gfjvu

Out[2]=2

ExpectedValue of a constant is the constant:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-jvs9l8

Out[1]=1

ExpectedValue of a random variable is the Mean:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-gt0wra

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-i9kg

Out[2]=2

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

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-feku3d

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-hksozh

Out[2]=2

ExpectedValue for a list is a Mean:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-ba2oj3

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-hemld8

Out[2]=2

CentralMoment is equivalent to an expected value:

In[1]:=1

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https://wolfram.com/xid/0rs07jqyi-mmuq7i

In[2]:=2

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https://wolfram.com/xid/0rs07jqyi-morekx

Out[2]=2

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