Question

X=reaction time, Y=length of life of electronic components Continuous joint probability distribution:

Compute: P(0 < X < 0.6 and 0.25 < Y < 0.5)

Answer 1

$f(x,y)=\left\{\begin{matrix} 4xy, 0<x<1.0, 0<y<1 & \\ 0, elsewhere & \end{matrix}\right.$

$P= \int \int F(x,y) dx dy$

P(0<x<0.5 + 0.25<y<0.5) fall in this region.

$P = \int^{0.6}_0 \int^{0.5}_{0.25} 4 xy dydx \\ P = \int^{0.6}_0 [4x (\frac{y^2}{2})]^{0.5}_{0.25} dx \\ = \int^{0.6}_0 2x [(\frac{1}{2})^2 -(\frac{1}{4})^2]dx \\ = (\frac{3}{8})\int^{0.6}_0 xdx \\ = \frac{3}{8}(\frac{x^2}{2})^{0.6}_0 \\ = \frac{3}{8}(0.6)^2 \\ = 0.0675$