Calculation of the probability of an event involving two non i.i.d. random variables Suppose $
\gamma _{D_{eq}}=\min \left\{ \gamma _R,\gamma _D \right\} ,\ \gamma _{E_{eq}}=\min \left\{ \gamma _R,\gamma _E \right\} $, where $\gamma_R$,   $\gamma_D$ and $\gamma_E$ are independent continuous random variables taking values in $(0,+\infty)$, and $\gamma_D,\gamma_E$ are identically distributed.
And $$P_{out}\left( C_{th} \right) =Pr\left\{ \log_2 \left( 1+\gamma _{D_{eq}} \ \right) -\log_2 \left( 1+\gamma _{E_{eq}} \ \right) \le C_{th} \right\} =Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}}+\varXi -1 \right\} 
\ge Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}} \ \right\} $$
$\\(\varXi=2^{C_{th}},\ C_{th}>0, $$\ C_{th}$ is a constant$)$.
I hope to express $Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}}\ \right\}$, the lower bound of $P_{out}\left( C_{th} \right)$, in terms of the probability distributions of $\gamma_R,\gamma_D,\gamma_E$ which are assumed to be known to us. As $\gamma_{D_{eq}}\ ,\gamma_{E_{eq}}$ are not independent according to their definition above, I cannot use this formula to calculate:
$$Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}} \ \right\} =\int_0^{\infty}{F_{\gamma _{D_{eq}}}\left( \varXi x \right) f_{\gamma _{E_{eq}}}\left( x \right) dx}$$
Instead, because $Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}} \ \right\} =Pr\left\{\min\left\{\gamma_D\ ,\gamma_R\right\}\le\varXi\min\left\{\gamma_E\ ,\gamma_R\right\}\right\}$, I consider separating the event $\left\{\min\left\{\gamma_D\ ,\gamma_R\right\}\le\varXi\min\left\{\gamma_E\ ,\gamma_R\right\}\right\}$ into disjoint events so that the result comes from the sum of several probabilities, i.e.
$$
Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}}\  \right\} =Pr\left\{ \gamma _D\le \gamma _R,\gamma _E\le \gamma _R,\gamma _D\le \varXi \gamma _E \right\} +Pr\left\{ \gamma _D\le \gamma _R,\gamma _E>\gamma _R \right\} 
$$
$$
+Pr\left\{ \gamma _D>\gamma _R,\gamma _E\le \gamma _R,\gamma _R\le \varXi \gamma _E \right\} +Pr\left\{ \gamma _D>\gamma _R,\gamma _E>\gamma _R \right\} 
$$
My question is how I could express these four probability terms and finally $Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}}\  \right\} $ through the CDF (cumulative distribution function) or PDF (probability density function) of $\gamma_R,\gamma_D,\gamma_E$. Actually I know they could be expressed by triple integrals of the multiplication of the PDFs of $\gamma_R,\gamma_D,\gamma_E$, but I cannot figure out the upper and lower limits of the triple integrals involved here. Moreover, as the PDF or CDF expressions of $\gamma_R,\gamma_D,\gamma_E$ are actually complicated and the triple integrals may be difficult to give closed-form expressions, I am also willing to know if there are any other methods to express $Pr\left\{ \gamma _{D_{eq}}\le \varXi \gamma _{E_{eq}}\  \right\} $ which may be  simpler.
Thank you for reading.
 A: In order to compute derivative quantities like the upper bound on $P_{out}$ as given above for the correlated quantities $Z_D=\min\{\gamma_R,\gamma_D\}~,~ Z_E=\min\{\gamma_R,\gamma_E\}$ one can calculate the joint distribution of these two variables with the help of the following definition in terms of the delta distribution:
$$\rho_2(z_D,z_E)=\int\delta(\min\{x_R,x_D\}-z_D)\delta(\min\{x_R,x_E\}-z_E)\rho(x_D,x_E,x_R)dx_Ddx_Edx_R$$
This assumes that the joint density of $\gamma_R, \gamma_E, 
\gamma_D$ is known. In the case at hand for example $\rho(x_D,x_E, x_R)=f(x_D)f(x_E)g(x_R)$ for some probability densities $f,g$. Then, by following the cumbersome casework carefully, one discovers that
$$\rho_2(z_D,z_E)=\delta(z_D-z_E)g(z_D)F^2(z_D)+\theta(z_D-z_E)f(z_E)[g(z_D)F(z_D)+G(z_D)f(z_D)]+\theta(z_E-z_D)f(z_D)[g(z_E)F(z_E)+G(z_E)f(z_E)]$$
where $F(x)=\int_x^\infty f(t)dt$. Armed with complete knowledge of the joint density one can also calculate the probability of the required event:
$$\text{Pr}(Z_D\leq \Xi Z_E)=\int_0^\infty \Xi dt\int_t^\infty d\eta~~\rho_2(\Xi t,\eta)$$
Each one of the four terms of $\rho_2$ after the double integration can be matched to one of the events' probabilities derived in the question above.
EDIT:
Steps to find $\rho_2$:

*

*Separate cases for the minimum functions that appear inside the delta distributions. There will be 4 terms in total and I will use Heaviside step functions again to express those:

$$\delta(\min\{x_R,x_D\}-z_D)\delta(\min\{x_R,x_E\}-z_E)=\theta(x_D-x_R)\theta(x_E-x_R)\delta(x_R-z_D)\delta(x_R-z_E)+\theta(x_R-x_D)\theta(x_E-x_R)\delta(x_D-z_D)\delta(x_R-z_E)+\theta(x_D-x_R)\theta(x_R-x_E)\delta(x_R-z_D)\delta(x_E-z_E)+\theta(x_R-x_D)\theta(x_R-x_E)\delta(x_D-z_D)\delta(x_E-z_E)$$


*Performing the trivial delta integrals for each term yields the integrals (summed in the order of the above equation)

\begin{align}\rho_2(z_D,z_E)=~&\delta(z_D-z_E)\int_{z_D}^\infty dx_D\int_{z_D}^\infty dx_E ~\rho(x_D,x_E,z_D)~+\theta(z_E-z_D)\int_{z_E}^\infty dx_E~\rho(z_D,x_E,z_E)~+\\&\theta(z_D-z_E)\int_{z_D}^\infty dx_D~\rho(x_D,z_E,z_D)~+\int_{\max\{z_D,z_E\}}^\infty dx_R~\rho(z_D,z_E,x_R)\end{align}
As promised, these four terms are in 1 to 1 match with the four terms presented in the question. Note here that the last term can also be written as
$$\int_{\max\{z_D,z_E\}}^\infty dx_R~\rho(z_D,z_E,x_R)=\theta(z_D-z_E)\int_{z_D}^\infty\rho(z_D,z_E, x_R) dx_R+ \theta(z_E-z_D)\int_{z_E}^\infty\rho(z_D,z_E, x_R) dx_R$$
which after substitution of the specific form for $\rho$ yields the final formula for $\rho_2$ above.
