Let $X$ and $Y$ be random variables with a joint probability density function given by: $f(x, y) =ke^{-(x+y)}$ for $0\leq x < \infty$ and $0\leq y <\infty$.
a) Find $k $.
b) Are $X$ and $Y$ independent?
To solve for $k$, would I just take the double integral of $e^{-(x+y)}$ both from 0 to infinity?
Not 100% sure if my approach to this is correct. Also how do I tell if $X$ and $Y$ are independent?
Any help is appreciated!
As regards a) you are right: we need that $$1=k\int_0^{+\infty}\int_0^{+\infty}e^{-(x+y)}dxdy=k\left(\int_0^{+\infty}e^{-x}dx\right)^2.$$ As regards b) first find the individual densities: $$f_X(x)=k\int_{y=0}^{+\infty}e^{-(x+y)}dy, \quad f_Y(y)=k\int_{x=0}^{+\infty}e^{-(x+y)}dx.$$ $X$ and $Y$ are independent iff $$f(x,y)=f_X(x)\cdot f_Y(y).$$