I'm interested in finding the CDF and PDF of $U_i$ defined as follows, 
$$U_i=\frac g{d^{\alpha}}$$
where $g$ is a gamma distributed random variable with shape $k$ and scale $\theta$, and $d$ is a random variable with distribution $f_d(x)=\frac {2x}{R^2}, 0\le x \le R$.  
To find the CDF of  $U_i$ 
$$F_{U_i}(y)=P(U_i<y)=\int_0^R F_g(yx^\alpha)dx$$ 
where $F_g(y)$ is the CDF of gamma distributed random variable. 
<br> After performing the integration, which requires the use of the expansion of incomplete gamma function, I came to the following CDF: 
$$F_{U_i}(y)=\sum_{n=0}^\infty \frac {2(-1)^nR^{k\alpha+n\alpha}y^{k+n}}{\Gamma(k)n!\theta^{k+n}(k+n)(k\alpha+n\alpha+2)}$$
I want the CDF in this format because my ultimate goal is to find the CDF of $U=\sum_{i=1}^NU_i$ which requires me to find the Laplace transform of $F_{U_i}(y)$ and raise it to power $N$. 
<br>Now it's easy to find the PDF of $U_i$, i.e. $f_{U_i}(y)=\frac {d}{dy}F_{U_i}(y)$. 
<br>I'm looking for someone to clarify these points to me: 
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My derived $f_{U_i}(y)$ doesn't make sense to me, because it's based on a divergent series. How can it's integration from zero to infinity equals one? 
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In Matlab I took the $K^{th}$ elements of the series, and plotted it along with the histogram of 1000 $U_i$'s, the two plots are not related. 
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If you look at the definition of $U_i$, $U_i$ can take values from 0 to infinity, high values take less probability. This fact is not clear in the derived distribution $f_{U_i}(y)$. What is the domain of the derived PDF?
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How can I simulate the derived PDF in Matlab to check if it's really correlated with $U_i$
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