Need a hint to start this problem for this distribution problem

Jul 2015
603
15
United States
food for thought, how would I compute just the density of X or Y on it's own? I get 1 for density of X, which I suspect is way off.
 
Jul 2015
603
15
United States
Density of X is 1 right? Density of Y is 1. What is the density of (-Y)? How do we calculate that?
 

romsek

MHF Helper
Nov 2013
6,725
3,030
California
I do these types of problems with finding $P[W < w]$ and differentiating.
 

romsek

MHF Helper
Nov 2013
6,725
3,030
California
are limits correct so far
Well... I think you'd benefit from plotting the integration regions for a few examples of $w$

say $-1 < w < 0,~w=0,~0 < w < 1$

The middle one is simple so you understand what's going on.
The outer two show the two different cases for integration limits.

Forget about convolution for the moment. While it's true that X+Y is distributed as the convolution of the X and Y distributions in practice (at least for me) it's rarely calculated that way.
 
Jul 2015
603
15
United States
I know this must be wrong..because my boundaries only has form of x=...., I know 1 integral has to have form of y=....
 

romsek

MHF Helper
Nov 2013
6,725
3,030
California
I'm getting that

$P[X-Y < w] = \begin{cases}\displaystyle \int_{-1/2}^{1/2+w} \int_{x-w}^{1/2} ~1 ~dy~dx &-1 \leq w < 0\\
\dfrac 1 2 &w=0 \\
\dfrac 1 2 \left(1-\displaystyle \int_{-1/2+w}^{1/2}\int_{-1/2}^{1/2-w}~1~dy~dx\right) &0 < w\leq 1
\end{cases}$

I'll let you do the algebra and differentiate to obtain the PDF.

Look at these at wolfram alpha
ContourPlot[ Boole[x - y < -0.75], {x, -1/2, 1/2}, {y, -1/2, 1/2}] ; (w=-0.75) don't include past the ;

ContourPlot[ Boole[x - y < 0], {x, -1/2, 1/2}, {y, -1/2, 1/2}] ; (w=0)

ContourPlot[ Boole[x - y < 0.75], {x, -1/2, 1/2}, {y, -1/2, 1/2}] ;(w=0.75)