| Topic: |
Science > Physics |
| User: |
"Mathematics Lover" |
| Date: |
09 Sep 2005 04:18:45 AM |
| Object: |
fractional moments of normally distributed random |
Let f be some positive fractional number (like 1/3 or 6/5).
Let x be a random number distributed according to the zero-mean normal
distribution with variance s^2.
i) What is E[ x^f ] ?
ii) What is E[ log(x) ] ? Does it exists in some sense?
Scoobie
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| User: "Robert Israel" |
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| Title: Re: fractional moments of normally distributed random |
09 Sep 2005 01:17:02 PM |
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In article <1126257524.990658.206540@g14g2000cwa.googlegroups.com>,
Mathematics Lover <markdemers15@hotmail.com> wrote:
Let f be some positive fractional number (like 1/3 or 6/5).
Let x be a random number distributed according to the zero-mean normal
distribution with variance s^2.
i) What is E[ x^f ] ?
According to Maple, Gamma((f+1)/2) 2^(f/2-1) s^f (1 + (-1)^f)/sqrt(pi).
As Julian mentioned, you need to choose which branch of x^f to use
for negative values of x (use the same branch for (-1)^f in the formula
above).
ii) What is E[ log(x) ] ? Does it exists in some sense?
Yes, with the same caveat about branches of log for negative numbers.
ln(s) + (-ln(2) - gamma + i pi)/2
if you use the principal branch of log.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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| User: "Julian V. Noble" |
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| Title: Re: fractional moments of normally distributed random |
09 Sep 2005 05:41:18 AM |
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Mathematics Lover wrote:
Let f be some positive fractional number (like 1/3 or 6/5).
Let x be a random number distributed according to the zero-mean normal
distribution with variance s^2.
i) What is E[ x^f ] ?
ii) What is E[ log(x) ] ? Does it exists in some sense?
Scoobie
You are looking for the integral
\int_{-\infty}^{\infty} { dx x^f \exp \left({-x^2}\right) }
in LaTeX notation. The problem is that z^f (z is a complex variable)
is a function with a branch point at z=0. If you cut the complex
plane from z = -\infty to 0 you can define z^f to be |z|^f for
z=x+i0, x > 0 (that is, on the positive real z-axis the function
is real and positive). On the negative axis, though, you must
specify whther you approach from below the axis or from above
the axis. That is, from above the axis you get
z^f -> |x|^f \exp\left({i f \pi}\right)
whereas from below you get
z^f -> |x|^f \exp\left({ -i f \pi}\right) ,
that is, the complex conjugate of the function.
To answer your question, it is possible to give a definite meaning
to the expectation of a function such as |x|^f but to define E( x^f )
requires some choices as to what you are trying to do.
BTW, E( |x|^f ) can be expressed as a complete Gamma function.
--
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
"For there was never yet philosopher that could endure the
toothache patiently."
-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
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| User: "" |
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| Title: Re: fractional moments of normally distributed random |
09 Sep 2005 06:46:07 AM |
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I understood there was to be no math in this course.
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| User: "Julian V. Noble" |
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| Title: Re: fractional moments of normally distributed random |
09 Sep 2005 12:43:57 PM |
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wrote:
I understood there was to be no math in this course.
The group is sci.math.symbolic..
^^^^
Seriously, if you expect a CAM program to do your math for you, you can
expect to make serious, possibly life-threatening mistakes. I don't mind
that much if they affect you only, but if you are designing a bridge,
vehicle or other widget used by the masses, even one mistake is too many.
In other words, you have to know what is inside the black box, else it
is too dangerous for you to use.
--
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
"Black boxes in engineering are like secrecy in government: better not
to trust what goes on out of sight."
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| User: "A N Niel" |
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| Title: Re: fractional moments of normally distributed random |
09 Sep 2005 07:42:43 AM |
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In article <1126266367.817599.151750@f14g2000cwb.googlegroups.com>,
<donstockbauer@hotmail.com> wrote:
I understood there was to be no math in this course.
This is not a course, this is three newsgroups.
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| User: "the softrat" |
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| Title: Re: fractional moments of normally distributed random |
09 Sep 2005 11:12:26 AM |
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On 9 Sep 2005 02:18:45 -0700, "Mathematics Lover"
<markdemers15@hotmail.com> wrote:
ii) What is E[ log(x) ] ? Does it exists in some sense?
Yes, over a non-infinite range.
the softrat
Sometimes I get so tired of the taste of my own toes.
mailto:softrat@pobox.com
--
Visualize using your turn signal.
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