Solutions - 5190-05

(a)

(1) Find the marginal PDF for $X$ by integrating the joint PDF with respect to $y$:

$$\begin{array}{c}{f}_X(x)={\int }_0^{\infty }2e^{-(x+2y)}dy\\ \\ \gg \gg \underset{b\to \infty }{\lim }{\int }_0^{b}2e^{-(x+2y)}dy\gg \gg -\underset{b\to \infty }{\lim }{\left[e^{-(x+2y)}\right]|}_0^b\gg \gg e^{-x}\end{array}$$

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(2) Find the marginal PDF for $Y$ by integrating the joint PDF with respect to $x$:

$$\begin{array}{c}{f}_Y(y)={\int }_0^{\infty }2e^{-(x+2y)}dx\\ \\ \gg \gg -\underset{b\to \infty }{\lim }{\left[2e^{-(x+2y)}\right]|}_0^b\gg \gg 2e^{-2y}\end{array}$$

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(b)

Integrate the joint PDF over the region above the line $y=x$:

$$\begin{array}{c}P[X>Y]={\int }_0^{\infty }{\int }_0^{x}2e^{-(x+2y)}dydx\\ \\ \gg \gg {\int }_0^{\infty }(e^{-x}-e^{-3x})dx\\ \\ \gg \gg {(-e^{-x}+\frac{1}{3}{e}^{-3x})|}_0^{\infty }\gg \gg 0-(-1+\frac{1}{3})\gg \gg \frac{2}{3}\end{array}$$