91Ó°ÊÓ

The function is $f(x,y)=C(y-x)e^{-y}-y<x<y,0<y<∞$

The joint density of X and Y is,

$f(x,y)=\left\{\begin{array}{cc}c(y-x)e^{-y}-y<x<y,0<y<∞& \\ 0& \text{Otherwise}\end{array}\right.$

The value of C is,

$C{∫}_{y=0}^{∞}{e}^{-y}\left({∫}_{x=-y}^y(y-x)dx\right)dy=1$

$C{∫}_{y=0}^{∞}{e}^{-y}{\left(xy-\frac{x^2}{2}\right)}_{x=-y}^{y}dy=1$

$C{∫}_{y=0}^{∞}{e}^{-y}\left(y(y-(-y\left)\right)-\frac{1}{2}\left(y^2-(-y{)}^2\right)\right)dy=1$

$C{∫}_{y=0}^{∞}{e}^{-y}\left(y(y+y)-\frac{1}{2}\left(y^2-y^2\right)\right)dy=1$

$C{∫}_{y=0}^{∞}{e}^{-y}\left(2y^2\right)dy=1$

$2C{∫}_{y=0}^{∞}{y}^2{e}^{-y}dy=1$

$$\begin{gathered} 2C\left[2\right]=1 \\ 4C=1 \\ C=\frac{1}{4} \end{gathered}$$

The function is$f(x,y)=C(y-x)e^{-y}-y<x<y,0<y<∞$.

The density function of X is,

$$\begin{gathered} f_x\left(x\right)=\frac{1}{4}{∫}_x^{∞}(y-x)e^{-y}dy \\ {f}_x\left(x\right)=\frac{1}{4}{∫}_x^{∞}\left(y{e}^{-y}-x{e}^{-y}\right)dy \\ =\frac{1}{4}\left\{{∫}_x^{∞}\left(y{e}^{-y}\right)dy-{∫}_x^{∞}\left(x{e}^{-y}\right)dy\right\} \\ =\frac{1}{4}\left\{{\left[-y{e}^{-y}-e^{-y}\right]}_x^{∞}+{\left[x{e}^{-y}\right]}_x^{∞}\right\} \\ =\frac{1}{4}\left\{{\left[-y{e}^{-y}-e^{-y}+x{e}^{-y}\right]}_x^{\mathrm{Î\pm }}\right\} \\ =\frac{1}{4}\left[-0+x{e}^{-x}-\left(0-e^{-x}\right)+x\left(0-e^{-x}\right)\right] \\ =\frac{e^{-x}}{4} \end{gathered}$$

The function is$f(x,y)=C(y-x)e^{-y}-y<x<y,0<y<∞$

The density function of Y is,

$$\begin{gathered} f_y\left(y\right)=\frac{1}{4}{e}^{-y}{∫}_{x=-y}^y(y-x)dx \\ =\frac{1}{4}{e}^{-y}{\left[yx-\frac{x^2}{2}\right]}_{x=-y}^y \\ =\frac{1}{4}{e}^{-y}\left[y(y-(-y\left)\right)-\frac{1}{2}\left(y^2-(-y{)}^2\right)\right] \\ =\frac{1}{4}{e}^{-y}\left[y(y+y)-\frac{1}{2}\left(y^2-y^2\right)\right] \\ =\frac{1}{2}{y}^2{e}^{-y} \end{gathered}$$

The function is$f(x,y)=C(y-x)e^{-y}-y<x<y,0<y<∞$

The value of $E\left[X\right]is,$

$$\begin{gathered} E\left[X\right]={∫}_{x=-∞}^{∞}xf\left(x\right)dx \\ =\frac{1}{4}\left[{∫}_{x=-∞}^{∞}x{e}^{-x}dx+{∫}_{-∞}^0\left(-2x^2{e}^x+x{e}^x\right)dx\right] \\ =\frac{1}{4}\left[{∫}_{x=-∞}^{∞}{x}^{2-1}{e}^{-x}dx-{∫}_0^{∞}\left(-2y^2{e}^{-y}+y{e}^{-y}\right)dy\right] \\ =\frac{1}{4}\left[\frac{\mathrm{Γ}\left(2\right)}{1}-{∫}_0^{∞}\left(2y^2{e}^{-y}\right)dy+{∫}_0^{∞}\left(y{e}^{-y}\right)dy\right] \\ =\frac{1}{4}\left[1-2{∫}_0^{∞}\left(y^{3-1}{e}^{-y}\right)dy-{∫}_0^{∞}\left(y^{2-1}{e}^{-y}\right)dy\right] \\ =\frac{1}{4}\left[1-2\left(\frac{\mathrm{Γ}\left(3\right)}{1}\right)-\frac{\mathrm{Γ}\left(2\right)}{1}\right] \\ =\frac{1}{4}[1-2(2!)-1!] \\ =-1 \end{gathered}$$

The function is$f(x,y)=C(y-x)e^{-y}-y<x<y,0<y<∞$

The value of $E\left[Y\right]$is,

$$\begin{gathered} E\left[Y\right]={∫}_{y=-∞}^{∞}yf\left(y\right)dy \\ =\frac{1}{2}\left[{∫}_0^{∞}y{y}^2{e}^{-y}dy\right] \\ =\frac{1}{2}\left[{∫}_0^{∞}{y}^{4-1}{e}^{-y}dy\right] \\ =\frac{1}{2}\left[\frac{\mathrm{Γ}\left(4\right)}{1^4}\right] \\ =\frac{1}{2}\left[3!\right] \\ =3 \end{gathered}$$