Q.8.14 - A First Course in Probability

Question

A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of these components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least .95? 

Step-by-Step Solution

Answer

 the number of components  of stocks required for given system is 

Step 1. Given information

From the given statements of the question, we have to find out the components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least .95.

Step 2. finding the probability limit

$$\begin{gathered} \mathrm{Let} {\mathrm{X}}_{\mathrm{i}}=\mathrm{life} \mathrm{time} \mathrm{of} {\mathrm{i}}^{\mathrm{th}} \mathrm{components} \\ \mathrm{given} \mathrm{that} \mathrm{mean} \mathrm{lifetime} \mathrm{\mu }=\mathrm{E}\left[{\mathrm{X}}_{\mathrm{i}}\right] \\ =100 \\ \mathrm{standard} \mathrm{deviation} =30 \mathrm{then} \mathrm{varience}( {\mathrm{\sigma }}^2) ={30}^2 \\ \mathrm{Take} \mathrm{X}=\mathrm{total} \mathrm{life} \mathrm{time} \mathrm{of} \mathrm{n} \mathrm{components} , \mathrm{so} \mathrm{X}=\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}} \end{gathered}$$

Based on the question lifetimes are independent random variables, so by using the properties of expectation we can write,

$$\begin{gathered} \mathrm{The} \mathrm{random} \mathrm{variable} \mathrm{X} \mathrm{with} \mathrm{mean} \mathrm{E}\left[\mathrm{x}\right]=\mathrm{E}\left[\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}\right] \\ =\underset{\mathrm{i}=1}{\overset{\mathrm{n}}{\sum \mathrm{\mu }}} \\ =\mathrm{n\mu } \\ =100\mathrm{n} \\ \mathrm{Var}\left(\mathrm{x}\right)=\mathrm{Var}\left(\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}\right)=\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{\sigma }}^2 \\ ={\mathrm{n\sigma }}^2 \\ ={30}^2{\mathrm{\sigma }}^2 \\ \mathrm{So} \mathrm{according} \mathrm{to} \mathrm{the} \mathrm{question} \mathrm{we} \mathrm{have} \mathrm{to} \mathrm{find} \mathrm{p}\left\{\mathrm{x}>2000\right\}>0.95 \end{gathered}$$

Step 3. Finding the number of components using the central limit theorem

The central limit theorem states that , states only that, for each a