Solutions - 5240-02

(a)

$$\begin{array}{c}P[X>Y]={\int }_0^{\infty }{\int }_y^{\infty }{f}_X(x)f_Y(y)dxdy\\ \\ \gg \gg \frac{1}{24}{\int }_0^{\infty }{\int }_y^{\infty }{e}^{-\frac{x}{3}}{e}^{-\frac{y}{8}}dxdy\\ \\ \gg \gg \frac{1}{24}{\int }_0^{\infty }{e}^{-\frac{y}{8}}\underset{b\to \infty }{\lim }{[-3e^{-\frac{x}{3}}]|}_y^{b}dy\\ \\ \gg \gg \frac{1}{8}{\int }_0^{\infty }{e}^{-\frac{y}{8}}{e}^{-\frac{y}{3}}dy\gg \gg \frac{1}{8}{\int }_0^{\infty }{e}^{-\frac{11y}{24}}dy\\ \\ \gg \gg \frac{3}{11}\underset{b\to \infty }{\lim }{[-e^{-\frac{11}{24}y}]|}_0^b\gg \gg \frac{3}{11}\end{array}$$

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(b)

Since $X\sim \text{Exp}\left(\frac{1}{3}\right)$ and $Y\sim \text{Exp}\left(\frac{1}{8}\right)$, by independence:

$$E[XY]=E[X]E[Y]=3\cdot 8=24$$

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(c)

Since $X$ and $Y$ are independent:

$$\text{Cov}[X,Y]=0$$

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(d)

Since $\text{Cov}[X,Y]=0$:

$$\rho [X,Y]=0$$