Calculating the statistical properties of the given PDF

[Arman777]

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

 

[mathman]

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 $=m_2-m_1^2$.

 

[BvU]

Any site or explanation

A textbook on statistics ?

 

[Arman777]

Okay I understand it. Thanks for the help

 

[Arman777]

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

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..

 

[BvU]

Okay It seems that I did not or at least I thought I was.
...
as I have said before.

Sorry I missed that -- 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:

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)

$$\phi(t)=\int_R e^{itx}f(x)dx$$

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 -- 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 ?)

$\$

 

[Arman777]

Not much in ordinary textbooks, I grant you.

Our textbook is really hard to understand since its not for starters..

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 $\phi(t)$

 

[Arman777]

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

 

[BvU]

A property of Fourier transforms is that the transform of a product is a convolution vice versa.

$\$

 

[Arman777]

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