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

$$P(X<0.5, Y<0.5)=\displaystyle\int_{0}^{0.5}\displaystyle\int_{0}^{0.5}\dfrac{3}{2}(x^2+y^2)dydx$$ $$=\dfrac{3}{2}\displaystyle\int_{0}^{0.5}[x^2y+\dfrac{y^3}{3}]dx\begin{matrix} 0.5 \\ 0 \end{matrix}$$ $$=\dfrac{3}{2}\displaystyle\int_{0}^{0.5}(\dfrac{1}{2}x^2+\dfrac{1}{24})dx$$ $$=\dfrac{3}{2}[\dfrac{1}{6}x^3+\dfrac{1}{24}x]\begin{matrix} 0.5 \\ 0 \end{matrix}$$ $$=\dfrac{3}{2}(\dfrac{1}{48}+\dfrac{1}{48})=\dfrac{1}{16}$$ $$P(X<0.5, Y<0.5)=\dfrac{1}{16}$$