I am stuck finding a marginal CDF of X given the random variables $X$ and $Y$ with the joint density:

$f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$


$f_X(x)=\int_x^{\infty} 2e^{-x-y} dy=2e^{-x}e^{-x}=2e^{-2x}$ for $0<x<\infty$. Also $F_X(x)=\int_0^{x} 2e^{-2t} dt =-e^{-2t}|_0^{x}=1-e^{-2x}, 0<x<\infty$.

  • $\begingroup$ Can you explain the second part ? $F_X(x)=1-e^{-2x}$ $\endgroup$ – i9-9980XE May 23 at 7:29
  • $\begingroup$ @i9-9980XE I have added some steps to my answer. $\endgroup$ – Kavi Rama Murthy May 23 at 7:31
  • $\begingroup$ Both of those answers constitute into the marginal CDF of X ? $\endgroup$ – i9-9980XE May 23 at 7:32
  • $\begingroup$ Yes, $f_X$ is the marginal density of $F$ and $F_X$ is the marginal distribution function of $X$. $\endgroup$ – Kavi Rama Murthy May 23 at 7:34

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