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Science Forum Index » Statistics - Math Forum » moment generating function...
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| minimus... |
 Posted: Sat May 17, 2008 3:49 pm |
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One can easily find a taylor series expansion's demostrative graph,
like the one at wikipedia. That is, how the taylor series expansion
approximates
another function is whon on a graph.
I was googling to find a similar graph for the moment generating function.
That is, the moment generating function approximating the distribution
function on a graph.
Do you know any book or ant pther type of source which has this graph? |
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| Herman Rubin... |
| Posted: Mon May 19, 2008 11:40 am |
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Guest
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In article <g0ngce$eb7$1 at (no spam) registered.motzarella.org>,
minimus <t24680 at (no spam) hotmail.com> wrote:
Quote: One can easily find a taylor series expansion's demostrative graph,
like the one at wikipedia. That is, how the taylor series expansion
approximates
another function is whon on a graph.
I was googling to find a similar graph for the moment generating function.
That is, the moment generating function approximating the distribution
function on a graph.
Do you know any book or ant pther type of source which has this graph?
The moment generating function is the Maclaurin (Taylor at 0)
expansion from the moments, if that converges. There is no
way to get a Taylor expansion of the distribution from the
moment generating function; however, if the characteristic
function has the analog of a moment generating function,
the density and the cdf can be calculated by a power series
using the "moments" of the characteristic function.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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| minimus... |
| Posted: Fri May 23, 2008 3:21 am |
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Guest
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Ok without reference to any 'expansion',
I am looking for a graphical representation of the moment generating
function.
because my book says that "the shape of the mgf in an open neighbourhood of
zero unuquly characterizes the
distribution of a random variable."
Quote: The moment generating function is the Maclaurin (Taylor at 0)
expansion from the moments, if that converges. There is no
way to get a Taylor expansion of the distribution from the
moment generating function; however, if the characteristic
function has the analog of a moment generating function,
the density and the cdf can be calculated by a power series
using the "moments" of the characteristic function.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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| Herman Rubin... |
| Posted: Fri May 23, 2008 4:30 pm |
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Guest
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In article <g15uqq$fqi$1 at (no spam) registered.motzarella.org>,
minimus <t24680 at (no spam) hotmail.com> wrote:
Quote: Ok without reference to any 'expansion',
I am looking for a graphical representation of the moment generating
function.
because my book says that "the shape of the mgf in an open neighbourhood of
zero unuquly characterizes the
distribution of a random variable."
The moment generating function can be looked at in two
ways; one is as sum m_k*t^k/k!, and the other as E(e^tx).
If the series converges for some non-zero t, the two are
equal. If the expected value of the exponential exists for
both a positive and a negative t, the series converges for
non-zero t.
From the fact that it exists in an interval, we can
conclude that it is what is called a real analytic function
in that interval. Now a real analytic function can be
continued to an analytic function of a complex variable
until it hits singularities. It cannot hit singularities
in the complex direction if it exists at the real part of
that complex number. The values of the function on the
line in the complex plane with a fixed real part are
sufficient to determine the distribution. One such case is
that of the real part 0, called the characteristic
function.
Quote: The moment generating function is the Maclaurin (Taylor at 0)
expansion from the moments, if that converges. There is no
way to get a Taylor expansion of the distribution from the
moment generating function; however, if the characteristic
function has the analog of a moment generating function,
the density and the cdf can be calculated by a power series
using the "moments" of the characteristic function.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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