Chapter 6: Q.6.45 (page 274)

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 $f (x) = x e^{- x}, x > 0,$ compute the density function of the length of time that the machine functions.

Short Answer

Expert verified

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

$\sum_{i = 3}^{5} (\begin{matrix} 5 \\ i \end{matrix}) ({(t + 1) e^{- t})}^{i} (1 - {(t + 1) e^{- t})}^{s - i}$

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,

$f (x) = \{\begin{cases} x e^{- x} x > 0 \\ 0 O t h e r w i s e \end{cases}$

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

$\int_{1}^{\infty} x e^{- x} d x {[- x e^{- x}]}_{t}^{x} - \int_{t}^{\infty} - e^{- x} d x t e^{- t} - {[e^{- x}]}_{t}^{\infty} t e^{- t} + e^{- t} e^{- t} (t + 1)$

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

localid="1647156973004" $\sum_{i = 3}^{5} (\begin{matrix} 5 \\ i \end{matrix}) ({(t + 1) e^{- t})}^{i} (1 - {(t + 1) e^{- t})}^{s - i}$

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Short Answer

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Complex machine :

Explanation :

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

Short Answer

Step by step solution

Complex machine :

Explanation :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Pure Maths

Statistics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.