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I have the following random variable ##\Gamma_m=\frac{a_m}{\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n+1}## where the random variables ##\{a_n\}## are independent and identically distributed random variables. The CDF of random variable ##a_n## if given by

[tex]F_{a_n}(x)=1-\frac{1}{1+x}[/tex]

Now I need to find the CDF of ##\Gamma_m##. I started like this:

[tex]F_m(\gamma)=Pr\left[\frac{a_m}{\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n+1}\leq \gamma\right]=1-Pr\left[\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n<\frac{a_m}{\gamma}-1\right]=1-\int_{a_m}Pr\left[\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n<\frac{a_m}{\gamma}-1\right]f_{a_m}(a_m)\,da_m[/tex]

where ##f_{a_m}(a_m)=1/(1+a_m)^2## is the PDF of the random variable ##a_m##. Apparently, ##Pr\left[\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n<\frac{a_m}{\gamma}-1\right]## is the CDF of the random variable ##\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n## for a given ##a_m##. How can I continue from here without resorting to Laplace or Fourier Transform? The reason why is that I need to write the CDF in terms of functions, because later I need to find its PDF.

Thanks

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# B How to find this CDF

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