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## Main Question or Discussion Point

Suppose I have the random variables ##Z_k=X_k/Y_k## with a PDF ##f_{Z_k}(z_k)## for ##k=1,\,2\,\ldots, K##, where ##\{X_k, Y_k\}## are i.i.d. random variables. I can find

[tex]\text{Pr}\left[\sum_{i=1}^3Z_k\leq \eta\right][/tex]

as

[tex]\int_{z_3=0}^{\eta}\int_{z_2=0}^{\eta-z_3}\int_{z_1=0}^{\eta-z_3-z_2}f_{Z_1}(z_1)f_{Z_2}(z_2)f_{Z_3}(z_3)\,dz_1dz_2dz_3[/tex]

Now suppose I arrange the random variables ##\{X_k\}_{k=1}^K## as

[tex]X_{(1)}\leq X_{(2)}\leq \cdots \leq X_{(K)}[/tex]

and define ##G_k=X_{(k)}/Y_{i_k}##, where ##i_k## is the index of the ##k^{\text{th}}## order statistic.

How can I find

[tex]\text{Pr}\left[\sum_{i=1}^3G_k\leq \eta\right][/tex]

[tex]\text{Pr}\left[\sum_{i=1}^3Z_k\leq \eta\right][/tex]

as

[tex]\int_{z_3=0}^{\eta}\int_{z_2=0}^{\eta-z_3}\int_{z_1=0}^{\eta-z_3-z_2}f_{Z_1}(z_1)f_{Z_2}(z_2)f_{Z_3}(z_3)\,dz_1dz_2dz_3[/tex]

Now suppose I arrange the random variables ##\{X_k\}_{k=1}^K## as

[tex]X_{(1)}\leq X_{(2)}\leq \cdots \leq X_{(K)}[/tex]

and define ##G_k=X_{(k)}/Y_{i_k}##, where ##i_k## is the index of the ##k^{\text{th}}## order statistic.

How can I find

[tex]\text{Pr}\left[\sum_{i=1}^3G_k\leq \eta\right][/tex]