Question

A complex machine is able to operate effectively as long as at least $3$of its $5$ motors are functioning. If each motor independently functions for a random amount of time with density function$f\left(x\right)=x{e}^{-x},x>0,$ compute the density function of the length of time that the machine functions.

Short answer

The probability that the machine will continue to work until at least time t is,

$\sum _{i=3}^5\left(\begin{array}{c}5\\ i\end{array}\right)({(t+1)e^{-t})}^i(1-{(t+1)e^{-t})}^{s-i}$

Step-by-step solution

Complex machine : 

A machine that combines two or more simple devices to make your task simpler.

Explanation :

A complicated machine can work efficiently if at least three of its five motors are operational. If each motor operates for a random period of time with a density function,

$f\left(x\right)=\left\{\begin{array}{l}x{e}^{-x}x>0\\ 0Otherwise\end{array}\right.$

The probability that a certain motor will function until at least time t if $t>0$. It is stated metaphorically as follows:

$$\begin{gathered} {\int }_1^{\infty }x{e}^{-x}dx \\ {[-x{e}^{-x}]}_t^x-{\int }_t^{\infty }-e^{-x}dx \\ t{e}^{-t}-{\left[e^{-x}\right]}_t^{\infty }t{e}^{-t}+e^{-t} \\ {e}^{-t}(t+1) \end{gathered}$$

Now calculate the probability that exactly one out of every five motors will operate until at least time t,

localid="1647156973004" $\sum _{i=3}^5\left(\begin{array}{c}5\\ i\end{array}\right)({(t+1)e^{-t})}^i(1-{(t+1)e^{-t})}^{s-i}$