# Determine $K$ and find the density functions of the random variables $Z = \max(X,Y)$ and $T = \min(X, Y )$.

The random variables $X$ and $Y$ have a joint density function given by; $$f(x,y)= \begin{cases} Kxy,\quad &0\leq x\leq 1,\; 0\leq y\leq 1,\\ 0,&\text{otherwise}. \end{cases}$$

Determine $K$ and find the density functions of the random variables $Z = \max(X,Y)$ and $T = \min(X, Y )$.

• This and your other recent questions do not fall under the category probability-theory. – Stefan Hansen Dec 15 '12 at 15:22

Recall that for some $f(x, y)$ to be a joint density function, two things must be satisfied:

1. $f(x, y)$ must be non-negative for all $x, y$.
2. The area under $f(x, y)$ must be 1.

The first part is easily satisfied by noting that $K$ must be non-negative.

The second part can be satisfied by setting the integral of $f(x, y)$ to 1:

$$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}f(x, y)\,\mathrm{d}x\mathrm{d}y = 1$$

Which can be simplified to:

$$\int\limits_{0}^{1}\int\limits_{0}^{1}Kxy\,\mathrm{d}x\mathrm{d}y = 1$$

At this point it shouldn't be difficult to solve for $K$.

For the $\max$ and $\min$ questions, you could observe that

$$\max(a, b) = \begin{cases}a\quad a\ge b,\\b\quad \mathrm{otherwise}.\end{cases}$$ $$\min(a, b) = \begin{cases}a\quad a\le b,\\b\quad \mathrm{otherwise}.\end{cases}$$

which ought to help.