I want to know that how conditioning will affect the CDF of dependent random variables. More specifically, let's suppose, $\Gamma_R={g\over A}$ and $\Gamma_D={g\cdot h\over B}$, where $g$ and $h$ are exponential random variables and $A$ and $B$ are some constant values. I want to find

$$\Pr \{min(\Gamma_R,\Gamma_D)<\gamma\}\quad\quad (1)$$

Since both $\Gamma_R$ and $\Gamma_D$ are statistically dependent because of the common term $g$.

So my question is: by conditioning on $g$, does the random variables $\Gamma_R$ and $\Gamma_D$ becomes independent? if yes then can I write eq.1 in the following form $$\Pr \{\Gamma_R<\gamma,\Gamma_D<\gamma\}\quad\quad (2)$$ $$\Pr \{g<\gamma A,h<\gamma{B\over g}\}\quad\quad (3)$$

$$=\int_0^\infty F_{\Gamma_R|g}(\gamma) \, \,\,\,\,F_{\Gamma_D|g}(\gamma)\, \,\,\,\,f_{g}(z) \, \,\,\,\,dz\quad\quad (4)$$

If yes then what will be the value of $F_{\Gamma_R|g}(\gamma)$, is it equals to 1? if yes, how? and can I just simply omit it from the above equation?

Any kind of help will be very much appreciated.


  • 1
    $\begingroup$ Are $A$ and $B$ constants or random variables? What is the joint distribution of $g$ and $h$? Conditioned on $g=g_0 > 0$, $\Gamma_R = \frac{g_0}{B}$ is a constant or a random variable depending on what $A$ is and so for constant $A$, the event $\{\Gamma_R < \gamma\}$ has conditional probability $0$ or $1$, and so on. $\endgroup$ – Dilip Sarwate May 30 '17 at 19:53
  • $\begingroup$ Thank you, Dilip for your kind reply, Yes, $A$ and $B$ are constants. $\endgroup$ – Baidal Kocham May 31 '17 at 0:53

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