Solutions - 5210-03

(1) Write the PDFs and joint PDF:

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:

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

---

(2) Find the CDF of $W=\text{Min}(X,Y)$:

$$\begin{array}{c}{F}_W(w)=P[W\le w]=P[X\le w\text{OR}Y\le w]\\ \\ \gg \gg 1-P[X\ge w\text{AND}Y\ge w]\\ \\ \gg \gg 1-{\iint }_{x,y\ge w}{f}_{X,Y}dA\end{array}$$

---

(3) Evaluate for $w\in [\mathrm{0,1}]$:

$$\begin{array}{c}{F}_W(w)=1-{\int }_w^1{\int }_w^{1}dxdy\\ \\ \gg \gg 1-{(1-w)}^2\gg \gg w(2-w)\end{array}$$

---

(4) Differentiate to find $f_W(w)$:

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