Suppose that the random variables X and Y have joint p.d.f.f(x,y)=\begin

Suppose that the random variables X and Y have joint p.d.f.

Evaluate the constant k.

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Given :

To find k
${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f\left(x,y\right)dxdy=1$
$={\int }_{0}^{2}{\int }_{-x}^{x}kx\left(x-y\right)dxdy=1$
$={\int }_{0}^{2}{\int }_{-x}^{x}k\left({x}^{2}-xy\right)dydx=1$
$={\int }_{0}^{2}k{\left[{x}^{2}y-\frac{x{y}^{2}}{2}\right]}_{-x}^{x}dx=1$
$={\int }_{0}^{2}k\left[{x}^{3}-\frac{{x}^{3}}{2}+{x}^{3}+\frac{{x}^{3}}{2}\right]dx=1$
$={\int }_{0}^{2}k\left[2{x}^{3}\right]dx=1$
$=2k{\left[\frac{{x}^{4}}{4}\right]}_{0}^{2}=1$
=2k(4)=1
$=k=\frac{1}{8}$