CDF of squared Euclidean Norm of Rayleigh random vector

Question

I have recently came across the following task:

Let us consider two vectors $h_1$ and $h_2$ such that $h_1$ is $1\times M$ and $h_2$ is $1 \times K$ vector. These vectors are assumed to be independent Rayleigh distributed random variables. Let $i \in \{1,2\}$. The PDF of $||h_i||^2$ is given as

$f_{||h_i||^2}(x) = \frac{x^{N-1}e^{-\frac{x}{\Omega_{h_i}}}}{\Gamma(N)\Omega^N_{h_i}}$---(1)

And its CDF is given as

$F_{||h_i||^2}(x) = 1-e^{-\frac{x}{\Omega_{h_i}}}\sum_{p = 0}^{P-1}\frac{x^p}{\Gamma(p+1)\Omega_{h_i}^p}$---(2)

My query is how this CDF is obtained from PDF of squared Euclidean Norm of Rayleigh random vector given in eq. (1)

Any help in this regard will be highly appreciated.

Answer 1

Suppose first that $\Omega=1$ just to eliminate clutter. Basically you are asked to find the indefinite integral $I= \int_x^{\infty} t^{n-1} e^{-t} dt$ which can be done by integration by parts using induction on $n$.

Alternatively evaluate $I$ by taking $(-\frac{d}{ds})^{n-1} \int_x^{\infty} e^{-st} dt$. and then setting $s=1$.