Question

Let X denote the diameter of an armores electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density f(x,y) ={ 1/y, 0<x<y<1, 0, elsewhere

Answer 1

We need the probability,

$p(X+Y\gt{1\over2})=1-p(X+Y\lt{1\over2})$

We have to find the limits of integration before finding this probability as follows,

Let $X+Y={1\over2}\implies Y={1\over2}-x$. The upper limit for Y is (1/2-x)

The upper limit for X can be determined by finding the point of intersection between the lines,

$y=x...(1)$

and

$y={1\over 2}-x...(2)$

Solving for x by substituting for y in equation (2) we have,

$x={1\over2}-x\implies x={1\over 4}$

The lower limits for X and Y are 0 and x respectively.

Now,

$1-p(X+Y\lt{1\over2})=1-\displaystyle\int^{1\over4}_0\displaystyle\int^{{1\over2}-x}_xf(x,y)dydx$

$=1-\displaystyle\int^{1\over4}_0\displaystyle\int^{{1\over2}-x}_x ({1\over y})dydx$

$=1-\displaystyle\int^{1\over 4}_0\lbrack ln({1\over 2}-x)-lnx\rbrack dx$

$=1+\lbrack({1\over 2}-x)ln({1\over 2}-x)-xlnx\rbrack|^{1\over4}_0$

$=1+{1\over 2}ln({1\over2})=0.6534$