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

$P(0<x<0.5,0<y<0.5)=\int^{0.5}_0\int^{0.5}_0 f(x)dxdy=$

$=\int^{0.5}_0\int^{0.5}_0 \frac{3}{2}(x^2+y^2)dxdy=\frac{3}{2}\int^{0.5}_0(x^3/3+xy^2)^{0.5}_0dy=$

$=\frac{3}{2}\int^{0.5}_0(1/24+y^2/2)dy=\frac{3}{2}(y/24+y^3/6)|^{0.5}_0=$

$=\frac{3}{2}(1/48+1/48)=1/16$