With multiple monitors, I have so far been dragging windows manually around. Is there a way to make a window open on a specific screen in a dual-head setup? (Not to be confused with multiple X server displays.)

Something like: SCREEN=2 firefox or open-in-screen 2 firefox.

If it matters, my environment is KDE using the KWin window manager. KWin window rules can only match some properties like title, but I could not find an environment variable as filter.

The secondary display extends the primary screen:

$ xrandr -q
Screen 0: minimum 320 x 200, current 3520 x 1080, maximum 32767 x 32767
LVDS1 connected 1600x900+0+0 (normal left inverted right x axis y axis) 382mm x 214mm
   1600x900       60.1*+
   1024x768       60.0  
   800x600        60.3     56.2  
   640x480        59.9  
VGA1 disconnected (normal left inverted right x axis y axis)
HDMI1 connected 1920x1080+1600+0 (normal left inverted right x axis y axis) 477mm x 268mm
   1920x1080      60.0*    50.0     59.9  
   1920x1080i     60.1     50.0     60.0  
   1680x1050      59.9  
   1280x1024      75.0     60.0  
   1440x900       59.9  
   1280x960       60.0  
   1280x720       60.0     50.0     59.9  
   1024x768       75.1     70.1     60.0  
   832x624        74.6  
   800x600        72.2     75.0     60.3     56.2  
   720x576        50.0  
   720x480        60.0     59.9  
   640x480        75.0     72.8     66.7     60.0     59.9  
   720x400        70.1  
DP1 disconnected (normal left inverted right x axis y axis)



Question: Probability density function with conditional distribution and correlation
Answer: On (c), we want to "integrate out" $y$. So the density function of $X$ is
$$\int_{y=x}^\infty 2e^{-x}e^{-y}\,dy.$$
Integration yields $2e^{-x}e^{-x}$. So $X$ has exponential distribution, parameter $2$.  

On (d), recall that the conditional density of $Y$ given that $X=x$ is
$$\frac{f_{X,Y}(x,y)}{f_X(x)}.$$
Note that the conditional density is $0$ for $y\lt x$.

On (e), the setup is good. There is a small sign error in the calculation. The number obtained is impossible, it is greater than $1$.

On (f), we will first need the covariance, that is, $E(XY)-E(X)E(Y)$. For that, we need to find 
$$\iint_D (xy) 2e^{-(x+y)}\,dydx,$$
where $D$ is the region where our joint density is positive.