Q.6.45 - A First Course in Probability

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. 

Step-by-Step Solution

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 1 : Complex machine :

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

Step 2 : 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,

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

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,

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