Solutions - 5210-02

(a)

(1) State the PDFs and CDFs:

$$f_X(x)=\{\begin{array}{cc}2e^{-2x}& x\ge 0\\ 0& \text{else}\end{array}{F}_X(x)=\{\begin{array}{cc}1-e^{-2x}& x\ge 0\\ 0& \text{else}\end{array}$$ $$f_Y(y)=\{\begin{array}{cc}3e^{-3y}& y\ge 0\\ 0& \text{else}\end{array}{F}_Y(y)=\{\begin{array}{cc}1-e^{-3y}& y\ge 0\\ 0& \text{else}\end{array}$$

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

Since $W=\text{Max}(X,Y)\ge X,Y$,

$$\begin{array}{c}{F}_W(w)=P[W\le w]=P[X\le w,Y\le w]\\ \\ \gg \gg P[X\le w]P[Y\le w]=F_X(w)F_Y(w)\end{array}$$ $$F_W(w)=\{\begin{array}{cc}1-e^{-2w}-e^{-3w}+e^{-5w}& w\ge 0\\ 0& \text{else}\end{array}$$

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(3) Differentiate to get the PDF:

$$f_W(w)=\{\begin{array}{cc}2e^{-2w}+3e^{-3w}-5e^{-5w}& w\ge 0\\ 0& \text{else}\end{array}$$

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(b)

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

$$\begin{array}{c}{F}_W(w)=P[W\le w]=1-P[W>w]\\ \\ \gg \gg 1-P[X>w,Y>w]=1-P[X>w]P[Y>w]\end{array}$$ $$F_W(w)=\{\begin{array}{cc}1-e^{-5w}& w\ge 0\\ 0& \text{else}\end{array}$$

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(2) Differentiate to get the PDF:

$$f_W(w)=\{\begin{array}{cc}5e^{-5w}& w\ge 0\\ 0& \text{else}\end{array}$$