Question

A device consists of a string of identical components connected in such way that the second component starts functioning as soon as the first component fails, the third component starts functioning as soon as the second component fails, etc. The lifetime of each component is mutually independent from the lifetime of any other component in the string. On average, it takes 12 years until the fourth component of such device fails. Find the probability that a device of this kind will be functioning for more than 15 years until its fourth component fails.

Answer 1

here expected time of failure for 4 components =12 years

therefore expected time of failure for a component =$\beta$ =12/4 =3 Years

and for gamma distribution $\alpha$ =4

hence for gamma distribution

probability that a device of this kind will be functioning for more than 15 years until its fourth component fails =P(X>15)=$\int_{15}^{\infty }$ f(x) dx= $\int_{15}^{\infty }$ $\frac{1}{3^{4}\tau (4)}x^{4-1}e^{-x/3}$ dx

=$\int_{15}^{\infty }$ (1/486)*x³ e^--x/3 dx

=(1/486)*(-3x³e^-x/3-27x²e^-x/3-162xe^-x/3-486e^-x/3)|^$\infty$₁₅ =0.2650