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I am doing a game theory question where $X$ is uniform over $[0,100]$ and $Y$ is uniform over $[10,110]$, where $X$ and $Y$ are the respective valuations of two people . I want to find the density function for $Z=X+Y$ in order to calculate some probabilities.

I've seen many solutions where X and Y follow a uniform distribution over $[0,1]$ but I don't seem to be able to replicate it using this range.

I have drawn a diagram and calculated some probabilities, e.g. $Pr(X+Y>60)=7/8$ but I would like to check them and calculate harder probabilities using the convolution of the two densities. Any help would be greatly appreciated!

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  • $\begingroup$ Hint: on the $xy$ grid $[0,100] \times [10,110]$, draw the level curves of $y=z-x$ for different $z$ values. This will tell you the regions to integrate over. The joint density will be $f_X(x)f_Y(y)=f_{X,Y}(x,y)$ by independence. $\endgroup$
    – ProfOak
    Mar 27, 2020 at 15:34
  • $\begingroup$ Intuitively for me the regions are $[0,10]$, $[10,200]$ and $[200,210]$ but why is it a joint density, I thought it would be a convolution of the x and y function (which I'm not sure how to compute) $\endgroup$ Mar 27, 2020 at 21:30

1 Answer 1

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Write $X=100A,\,Y=10+100B,\,C=A+B$ so $Z=10+100C$. Then $C$ has an $n=2$ Irwin–Hall distribution, which can be obtained as per @zugzug's comment. In terms of Iverson brackets$$f_C(c)=|1-c|[c\in[0,\,2]],\,f_Z(z)=\frac{1}{100}|1-(z-10)/100|[z\in[10,\,210]].$$

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