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Let $X$, $Y$ be independent random variables with the common pdf \begin{eqnarray*} f(u) &=& \left\{\begin{array}{ll} u\over2 & \mbox{for } 0 < u < 2\\ 0 &\mbox{elsewhere} \end{array}\right.\\ \end{eqnarray*} Set up an explicit double integral for $P(X Y > 1)$

Let $Z$ be the maximum of $X,Y$ (That is, $Z = X$ if $X \geqslant Y$, and $Z= Y$ if $Y > X$). Find $P(Z\leqslant 1)$

Find the pdf $g(z)$ of $Z$, being sure to define $g(z)$ for all numbers $z$.

This is a problem on a practice exam I'm studying, but I really have no idea how to approach the problem.

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up vote 1 down vote accepted
  • Sketch the plane with coordinate axes and draw on it the square region (of area $4$) over which the joint density is nonzero. Determine the joint density function $f_{X,Y}(u,v) = f_X(u)f_Y(v)$.

  • Sketch the hyperbola $xy=1$ in the first quadrant and persuade yourself $P\{XY > 1\}$ is the total probability mass in one of the two regions into which the hyperbola divides the square.

  • Find $P\{XY>1\}$ by integrating $f_{X,Y}(u,v)$ over the region you just identified.

$F_Z(z) = P\{Z \leq z\} = P\{X \leq z, Y \leq z\} = P\{X \leq z\}P\{Y \leq z\}$.

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