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MomentGeneratingFunction
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MomentGeneratingFunction

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gives the moment-generating function for the distribution dist as a function of the variable t.

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MomentGeneratingFunction[dist,{t1,t2,…}]

gives the moment-generating function for the multivariate distribution dist as a function of the variables t1, t2, … .

Details

Examples

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

Compute the moment-generating function (mgf) for a continuous univariate distribution:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-jadjlh
Out[1]=1

The mgf for a univariate discrete distribution:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-gsgux3
Out[1]=1

The mgf for a multivariate distribution:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-c22tur
Out[1]=1

Scope  (5)Survey of the scope of standard use cases

Compute the moment-generating function (mgf) for a formula distribution:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-dpdnhx
Out[1]=1

Find the mgf for a function of a random variate:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-5p9gg
Out[1]=1

Find the mgf for a data distribution:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-gj1haf
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-o667i
Out[2]=2

Compute the mgf for a censored distribution:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-hznr
Out[1]=1

Find the mgf for the slice distribution of a random process:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-hha5jz
Out[1]=1

Applications  (3)Sample problems that can be solved with this function

Find the moment-generating function of the sum of random variates:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-lfhsat
Out[1]=1

Check that it is equal to the product of generating functions:

In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-les66n
Out[2]=2

When it coincides with the mgf of BinomialDistribution:

In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-dr447n
Out[3]=3

Confirm with TransformedDistribution:

In[4]:=4
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https://wolfram.com/xid/0cprjjr8m76xzm-k32c8
Out[4]=4

Reconstruct the PDF of a positive real random variate from its moment-generating function:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-hf7fvr
In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-cew0f
Out[2]=2

Check the result:

In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-df39n5
Out[3]=3

Illustrate the central limit theorem on the example of PoissonDistribution:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-6p42

Find the moment-generating function for the standardized random variate:

In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-divtde
Out[2]=2

Find the moment-generating function for the sum of standardized random variates rescaled by :

In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-bpqcbx
Out[3]=3

Find the large limit:

In[4]:=4
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https://wolfram.com/xid/0cprjjr8m76xzm-ggbzom
Out[4]=4

Compare with the moment-generating function of a standard normal distribution:

In[5]:=5
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https://wolfram.com/xid/0cprjjr8m76xzm-h6yp8
Out[5]=5

Properties & Relations  (5)Properties of the function, and connections to other functions

MomentGeneratingFunction is equivalent to Expectation of :

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-dxjpgc
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-bto8oy
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-6ben4b
Out[3]=3

MomentGeneratingFunction is an exponential generating function for the sequence of moments:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-ep1jwd
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-e8ssdk
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-ipi3bn
Out[3]=3

Use SeriesCoefficient to find moment :

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-dcq08u
Out[1]=1

Use Moment directly:

In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-pifvvp
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-pyvump
Out[3]=3

MomentGeneratingFunction is a LaplaceTransform for positive random variables:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-g58jmc
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-dngxfc
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-4oo5f9
Out[3]=3

MomentGeneratingFunction is a ZTransform for discrete positive random variates:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-d7tbia
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-ptor8
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-0t1fos
Out[3]=3

Possible Issues  (2)Common pitfalls and unexpected behavior

For some distributions with long tails, moments of only several low orders are defined:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-cqpy04
Out[1]=1

Correspondingly, MomentGeneratingFunction is undefined:

In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-dv1sm4
Out[2]=2

Analytic continuation of CharacteristicFunction can sometimes be defined:

In[3]:=3
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https://wolfram.com/xid/0cprjjr8m76xzm-hepruo
Out[3]=3

MomentGeneratingFunction is not always known in closed form:

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-b2h9gq
Out[1]=1

Use Moment to evaluate particular moments:

In[2]:=2
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https://wolfram.com/xid/0cprjjr8m76xzm-b6lvfl
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-xhy7g
In[1]:=1
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https://wolfram.com/xid/0cprjjr8m76xzm-iiy93r
Out[1]=1
Wolfram Research (2010), MomentGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html.
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Wolfram Research (2010), MomentGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html.

Text

Wolfram Research (2010), MomentGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html.

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Wolfram Research (2010), MomentGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html.

CMS

Wolfram Language. 2010. "MomentGeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html.

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Wolfram Language. 2010. "MomentGeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html.

APA

Wolfram Language. (2010). MomentGeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html

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Wolfram Language. (2010). MomentGeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html

BibTeX

@misc{reference.wolfram_2025_momentgeneratingfunction, author="Wolfram Research", title="{MomentGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}", note=[Accessed: 01-April-2025 ]}

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@misc{reference.wolfram_2025_momentgeneratingfunction, author="Wolfram Research", title="{MomentGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}", note=[Accessed: 01-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_momentgeneratingfunction, organization={Wolfram Research}, title={MomentGeneratingFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}, note=[Accessed: 01-April-2025 ]}

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@online{reference.wolfram_2025_momentgeneratingfunction, organization={Wolfram Research}, title={MomentGeneratingFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}, note=[Accessed: 01-April-2025 ]}