A Calculating the statistical properties of the given PDF

Arman777
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For instance if we are given only a PDF in the form of ##p(x)##, how can one calculate the characteristic function, the mean, and the variance of these PDF's ?

Any site or explanation will be enough for me
 
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There are standard formulas given pdf ##f(x)##. Char. fcn. ##\phi(t)=\int_R e^{itx}f(x)dx##, moments ##m_k=\int_R x^kf(x)dx##, variance ##=m_2-m_1^2##.
 
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Arman777 said:
Any site or explanation
A textbook on statistics ?
 
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Okay I understand it. Thanks for the help
 
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Okay It seems that I did not or at least I thought I was. Let me take a uniform distribution in the form of ##p(x) = 1/2b##. The characteristic function is given by

$$p(k) = \frac{ie^{-ikx}}{2ak}$$

From here I want to calculate the mean and the variance as I have said before. I want to use this equation

1618137328629.png


so I get

$$ln(\frac{ie^{-ikx}}{2ak}) = ik<x>_c - \frac{k<x^2>_c}{2}$$

$$ln(\frac{i}{2ak})-ikx = ik<x>_c - \frac{k<x^2>_c}{2}$$

but from here I don't know what to do..
 
Arman777 said:
Okay It seems that I did not or at least I thought I was.
...
as I have said before.
Sorry I missed that :wink: -- can't make much sense of the first one and don't believe the second

Interesting thing, this FT aspect of a pdf. Not much in ordinary textbooks, I grant you.I will switch to 'shut up and learn mode' after these comments:
Arman777 said:
The characteristic function is given by
$$p(k) = \frac{ie^{-ikx}}{2ak}$$
This can't be right:
##p(x) = 1/2b## suggests a uniform distribution from ##-b## to ##+b##
So I would expect (using ##\phi(t)##, not ##p(k)## which is confusing)
mathman said:
$$\phi(t)=\int_R e^{itx}f(x)dx$$
something that depends on ##k## and ##b##, but not on ##x## ! (Lazy me: ##\displaystyle{\sin bt\over bt} ##, which I now 'all of a sudden' recognize and remember :cool: -- from the FT world)

By the same token the ##\int x^n p(x)## goodies you want to derive from ##\ \phi(k)\ ## should be convolutions in the ##t## domain (right ?)

##\ ##
 
BvU said:
Not much in ordinary textbooks, I grant you.
Our textbook is really hard to understand since its not for starters..
BvU said:
p(x)=1/2b suggests a uniform distribution from −b to
Yes my mistake, sorry about that. But in our book the notation is p(k) so I cannot use ##\phi(t)##
 
BvU said:
By the same token the ∫xnp(x) goodies you want to derive from ϕ(k) should be convolutions in the t domain (right ?)
I don't know what this means
 
A property of Fourier transforms is that the transform of a product is a convolution vice versa.

##\ ##
 
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Okay, this time I really solved the problem. Thanks for the help.
 
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