0
$\begingroup$

I have a Non-central $\chi^2$ distribution with the following PDF

$$ f(t) = \frac{K+1}{F}\cdot\exp\Bigg({\frac{-KF - (K+1)t}{F} I_o\bigg(2 \sqrt{\frac{K(K+1)t}{F}}\bigg)}\Bigg)$$

K,F are constants, while $t$ is the fading channel power.

Where $I_o$ is the modified Bessel function of Zeroth-Order of the First kind.

How can I derive the CDF of the above equation? I am trying to find P[t < H] actually.

$\endgroup$
  • $\begingroup$ Integration??? Whether or not a closed form solution exists is a separate question. $\endgroup$ – wolfies Apr 3 '18 at 13:18
0
$\begingroup$

Central chi-squared distribution with $\nu$ degrees of freedom, that is$\chi^2_\nu$ ($\nu \in\mathbb N$), only has an elementary closed form CDF for each even value of $\nu$, and that doesn't even mean that a generic elementary expression can be found involving all cases, but that an elementary expression exists for each even $\nu$. In any case, there's not an elementary expression for odd values of $\nu$. This is the case for central chi-squared, let alone the more generic non-central chi-squared.

Anyway, closed form is always an ambiguous concept. For instance, using incomplete gamma function there is a closed expression in the central case (and maybe in the non-central case too) but this is almost as general as using the very same definition of CDF (with an integral from $-\infty$ to a certain point) as such a closed form expression.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.