   Chapter 13.7, Problem 43E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Repair time In a manufacturing process involving several machines, the average down time t (in hours) for a machine that needs repair has the probability density function f ( t ) = 0.5 e − 0.5 t   t ≥ 0 Find the probability that a failed machine’s down time is (a) 2 hours or more. (b) 8 hours or more.

(a)

To determine

To calculate: The probability that a failed machine’s down time is 2 hours or more if the average down time t (in hours) for a machine that requires repair has the probability density function f(t)=0.5e0.5t.

Explanation

Given Information:

The average down time t (in hours) for a machine that requires repair has the probability density function,

f(t)=0.5e0.5t

Formula used:

When f(x) is a continuous probability density function, then

Pr(axb)=abf(x)dx

According to the exponential rule of integrals,

exdx=ex+C

Calculation:

It is provided that the average down time t (in hours) for a machine that requires repair has the probability density function,

f(t)=0.5e0.5t

Now, the formula for probability for continuous probability density function is,

Pr(axb)=abf(x)dx

Thus, to obtain the probability that a failed machine’s down time is 2 hours or more, substitute 2 for a, 0.5e0.5t for f(x) and for b in above formula to get,

P(x2)=20.5e0.5tdx

Now, use the exponential rule of integrals to obtain,

P(x2)=20

(b)

To determine

To calculate: The probability that a failed machine’s down time is 8 hours or more if the average down time t (in hours) for a machine that needs requires has the probability density function f(t)=0.5e0.5t.

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