Question

A production facility contains two machines that are used to rework items that are initially defective. Let 𝑋 be the number of hours that the first machine is in use and let 𝑌 be the number of hours that the second machine is in use, on a randomly chosen day. Assume that 𝑋 and 𝑌 have a joint probability density function given by 𝑓(𝑥) = { 3 2 (𝑥 2 + 𝑦 2 ) 0 < 𝑥 < 1 𝑎𝑛𝑑 0 < 𝑦 < 1 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. a. What is the probability that both machines are in operation for less than half an hour?

Answer 1

To find the probability that both machines are in operation for less than half an hour, we determine the probability,

$p(0\lt x\lt 0.5,0\lt y\lt 0.5)=\displaystyle\int^{0.5}_0\displaystyle\int^{0.5}_0 f(x,y)dydx\\ =\displaystyle\int^{0.5}_0\displaystyle\int^{0.5}_0 {3\over2}(x^2+y^2)dydx\\ ={3\over2}\displaystyle\int^{0.5}_0\displaystyle\int^{0.5}_0 (x^2+y^2)dydx ={3\over2}\displaystyle\int^{0.5}_0(x^2y+{y^3\over3})|^{0.5}_0dx\\ ={3\over2}\displaystyle\int^{0.5}_0(0.5x^2+{1\over24})dx={3\over2}({1\over6}x^3+{x\over24})|^{0.5}_0={3\over2}({1\over48}+{1\over48})={1\over16}$

Therefore, the probability that both machines are in operation for less than half an hour is ${1\over16}$