Question

Let X denote the diameter of an armored 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 2. Suppose that X and Y have the joint density Ky 0<<y< 2; f(x,y) = 0 otherwise. {K 1. Determine the value of the constant K. 2. Determine the P(X+Y > 0.5).

Answer 1

1) Since total probability is 1 so

f(x, y)ddy = 1 0 ry Kyddy = 1 [ Kydy = 1 2 3 2 K = 1 3 0 K 0 co

2) The region is shown below

3 2.5 2 1.5- 1、 0.5 81 -1.5 -1 -0.5 0.5、 1.5 2 2.5 -0.5 -1 -1.5

Hence the required probability is

$p=\int_{0}^{0.25}\int_{0.5-x}^{2} f(x,y)dydx+ \int_{0.25}^{2}\int_{ x}^{2} f(x,y)dydx\\ \\~~ ~~~~~={3\over 8}\left ( \int_{0}^{0.25}\int_{0.5-x}^{2} y dydx+ \int_{0.25}^{2}\int_{ x}^{2} ydydx\right )\\\\ ~~~~~~~={3\over 8}\left ( \int_{0}^{0.25}\left [ y^2\over 2 \right ]_{0.5-x}^{2} dx+ \int_{0.25}^{2}\left [ y^2\over 2 \right ]_{ x}^{2} dx\right ) \\ \\ ~~~~~~~={3\over 16}\left ( \int_{0}^{0.25}[4-(0.5-x)^2]dx+ \int_{0.25}^{2}[4-x^2] dx\right ) \\ ~~~~~~~={3\over 16}[0.96354+4.33854] \\ ~~~~~~~\approx 0.99414$