Solutions - 5220-05

(1) Write PDFs:

PDFs of $X$ and $Y$:

$$f_X(x)=\{\begin{array}{cc}1& 0\le x\le 1\\ 0& \text{otherwise}\end{array}{f}_Y(y)=\{\begin{array}{cc}1& 0\le y\le 1\\ 0& \text{otherwise}\end{array}$$

Joint PDF using independence:

$$f_{X,Y}(x,y)=\{\begin{array}{cc}1& x,y\in [\mathrm{0,1}]\\ 0& \text{otherwise}\end{array}$$

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Option 1: CDF first

(2) CDF of sum:

Write $W=X+Y$. Then:

$$\begin{array}{c}{F}_W(w)=P[X+Y\le w]\\ \\ \gg \gg {\iint }_{x+y\le w}{f}_{X,Y}dA\gg \gg \underset{\begin{array}{c}x+y\le w\\ x,y\in [\mathrm{0,1}]\end{array}}{\iint }1\cdot 1dydx\end{array}$$

When $0\le w<1$:

$$F_W(w)={\int }_{x=0}^w{\int }_{y=0}^{w-x}dydx\gg \gg \frac{w^2}{2}$$

When $1\le w<2$:

$$F_W(w)=1-{\int }_{x=w-1}^1{\int }_{y=w-x}^{1}dydx\gg \gg -w^2/2+2w-1$$

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(3) PDF from CDF:

$$f_W(w)=\{\begin{array}{cc}w& w\in [\mathrm{0,1}]\\ -w+2& w\in [\mathrm{1,2}]\\ 0& \text{otherwise}\end{array}$$

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Option 2: Convolution formula

(2) Convolution:

$$f_W(w)={\int }_{-\infty }^{+\infty }{f}_{X,Y}(u,w-u)du$$

For $w\in [\mathrm{0,1}]$:

$$f_W(w)={\int }_0^{w}du\gg \gg w$$

For $w\in [\mathrm{1,2}]$:

$$\begin{array}{c}{f}_W(w)={\int }_{w-1}^{1}du\gg \gg 1-(w-1)\\ \\ \gg \gg -w+2\end{array}$$