91影视

Random Variables $X,Y$

Density function $f(x,y)=\left\{\begin{array}{l}2e^{鈭�2x}/x鈥呪赌呪赌呪赌�0鈮�x<\mathrm{鈭�},0鈮�y鈮�x\\ 0鈥呪赌呪赌呪赌�\text{otherwise}\end{array}\right.$

$Cov(X,Y)=?$

From the information, observe that the random variable $X$and $Y$are the random variables that have the joint probability density function is as follows:

$f(x,y)=\left\{\begin{array}{l}2e^{鈭�2x}/x鈥呪赌呪赌呪赌�0鈮�x<\mathrm{鈭�},0鈮�y鈮�x\\ 0鈥呪赌呪赌呪赌�\text{otherwise}\end{array}\right.$

Calculate$E\left(XY\right)$

$E\left(XY\right)={鈭�}_0^{鈭�}{鈭�}_0^{x}xyf(x,y)dydx$

$={鈭�}_0^{鈭�}{鈭�}_0^{x}xy\frac{2}{x}{e}^{-2x}dydx$

$={鈭�}_0^{鈭�}{鈭�}_0^{x}2y{e}^{-2x}dydx$

$=2{鈭�}_0^{鈭�}{e}^{-2x}{\left(\frac{y^2}{2}\right)}_0^{x}dx$

By integrating the function,

$={鈭�}_0^{鈭�}{e}^{-2x}\left(x^2\right)dx$

$={鈭�}_0^{鈭�}{x}^2{e}^{-2x}dx$

Integration by parts:

$={\left(x^2\frac{e^{-2x}}{-2}\right)}_0^{鈭�}-{鈭�}_0^{鈭�}2x\left(\frac{e^{-2x}}{-2}\right)dx$

$=-\frac{1}{2}\left(鈭�-0脳e^{-0}\right)+{鈭�}_0^{鈭�}x{e}^{-2x}dx$

$=0+{鈭�}_0^{鈭�}x{e}^{-2x}dx$

$={\left(x\frac{e^{-2x}}{-2}\right)}_0^{鈭�}-{鈭�}_0^{鈭�}\frac{e^{-2x}}{-2}dx$$={\left(x\frac{e^{-2x}}{-2}\right)}_0^{鈭�}-{鈭�}_0^{鈭�}\frac{e^{-2x}}{-2}dx$

$=0+\frac{1}{2}{鈭�}_0^{鈭�}{e}^{-2x}dx$

$=\frac{1}{2}{\left[\frac{e^{-2x}}{-2}\right]}_0^{鈭�}$

$=-\frac{1}{4}\left[e^{-鈭�}-e^{-0}\right]$

$=-\frac{1}{4}(0-1)$

$=\frac{1}{4}\left(1\right)$

$=\frac{1}{4}$

Calculate $E\left(X\right)$

$E\left(X\right)={鈭�}_0^{鈭�}{鈭�}_0^{x}xf(x,y)dydx$

$={鈭�}_0^{鈭�}{鈭�}_0^{x}x\frac{2}{x}{e}^{-2x}dydx$

$={鈭�}_0^{鈭�}{鈭�}_0^{x}2e^{-2x}dydx$

$=2{鈭�}_0^{鈭�}{e}^{-2x}(y{)}_0^{x}dx$

$=2{鈭�}_0^{鈭�}{e}^{-2x}(x-0)dx$

$=2{鈭�}_0^{鈭�}x{e}^{-2x}dx$

$=2\left[{\left(x\frac{e^{-2x}}{-2}\right)}_0^{鈭�}-{鈭�}_0^{鈭�}{e}^{-2x}dx\right]$

$=\left[0-{\left(\frac{e^{-2x}}{-2}\right)}_0^{鈭�}\right]$

$=-\frac{1}{2}\left(e^{-鈭�}-e^{-0}\right)$

$=-\frac{1}{2}(0-1)$

$=\frac{1}{2}\left(1\right)$

$=\frac{1}{2}$

Calculate $E\left(Y\right)$

$E\left(Y\right)={鈭�}_0^{鈭�}{鈭�}_0^{x}yf(x,y)dydx$

$={鈭�}_0^{鈭�}{鈭�}_0^{x}y\frac{2}{x}{e}^{-2x}dydx$

$={鈭�}_0^{鈭�}{鈭�}_0^x\frac{2}{x}{e}^{-2x}\left(ydy\right)dx$

$={鈭�}_0^{鈭�}\frac{2}{x}{e}^{-2x}{\left(\frac{y^2}{2}\right)}_0^{x}dx$

$={鈭�}_0^{鈭�}\frac{1}{x}{e}^{-2x}\left(x^2\right)dx$

$={鈭�}_0^{鈭�}x{e}^{-2x}dx$

$=\left[{\left(x\frac{e^{-2x}}{-2}\right)}_0^{鈭�}-{鈭�}_0^{鈭�}\frac{e^{-2x}}{-2}dx\right]$

$=\left[0+\frac{1}{2}{\left(\frac{e^{-2x}}{-2}\right)}_0^{鈭�}\right]$

$=-\frac{1}{4}\left(e^{-鈭�}-e^{-0}\right)$

$=-\frac{1}{4}(0-1)$

$=\frac{1}{4}\left(1\right)$

$=\frac{1}{4}$

Therefore, the calculation of $Cov(X,Y)$

$Cov(X,Y)=E(XY)-E\left(X\right)E\left(Y\right)$

$=\frac{1}{4}-\frac{1}{2}脳\frac{1}{4}$

$=\frac{1}{4}-\frac{1}{8}$

$=\frac{2-1}{8}$

$=\frac{1}{8}$

Hence, the computation of$Cov(X,Y)$is$\frac{1}{8}.$