# Integration, Marginalization

• touqra
In summary, the conversation discusses the probability density function for one axis given an exponential decay of a material in two-dimensional space. It involves marginalizing and solving integrations using Mathematica and Matlab, but with limited success due to Bessel functions. A possible solution is then proposed using the MacDonald function and a Fourier transform.

#### touqra

I'll like to know the probability density function for one of the x or y axis, given that there is an exponential decay of a material in two-dimensional space. So, that means I have to marginalize, say y and keep x, but I couldn't solve the integration. I even tried with Mathematica and Matlab. Mathematica couldn't solve it. Matlab gives a Bessel function when x == 1, but when x != 1, it couldn't solve it. Please help.

$$PDF(x) = \int e^{-r} \, dy = \int e^{-\sqrt{x^2+y^2}} \, dy$$

Something like ## K_0(|x|)##, where K is the MacDonald function.
so $$f(x_0)=\int \exp(-r) dy|_{x=x_0}=\int dx dy \exp(-r) \delta(r \cos \phi -x_0)=$$
$$=\int dr d\phi r \exp(-r) \frac{1}{2\pi} \int dk \exp(ik (r \cos \phi -x))$$
Now we first integrate over ##\phi## using ##\int d\phi \exp(ikr\cos\phi)=2\pi J_0(kr)##:
$$f(x)=\frac{1}{2\pi}\int dk \int dr r \exp(-r)2 \pi J_0(kr)\exp(ikx_0)$$
The integral over r I found in Magnus Oberhettinger, Formeln und Saetze fuer die speziellen Funktionen der Physik, p33
##\int_0^\infty \exp(-at)J_\nu(bt)t^\nu dt=\frac{(2b)^\nu \Gamma(\nu+1/2)}{(a^2+b^2)^{\nu+1/2} \sqrt{\pi}}##
so that (setting x_0=x)
$$f(x)= \int dk (k^2+1)^{-1/2} \exp(ikx)$$
The Fourier transform is standard and yields the MacDonald function.

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Integration refers to the process of bringing different groups of people together and promoting their equal participation and inclusion in society. It involves breaking down barriers and creating opportunities for diverse individuals to interact, learn from each other, and live together harmoniously.

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