# Need a hint to start this problem for this distribution problem

#### math951

Am I on the right path to solving this? Are my boundaries right for the double integral?

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#### math951

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.

#### math951

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
I do these types of problems with finding $P[W < w]$ and differentiating.

#### math951

are limits correct so far

#### romsek

MHF Helper
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.

#### math951

I feel I am messing up on analysis of boundaries.

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#### math951

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
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)