Exponential distribution.

May 2010
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the time to failure of a machine when used for 1 hour can be modeled by a exponential distribution with Beta value = 50.

Find the probability that the machine will not fail during 5 consecutive 8 hour shifts.


In this modified case should i take Beta value as = 50 * 8 * 5 ?

Please help me.
 
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galactus

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The exponential distribution can be written as \(\displaystyle {\lambda}=\frac{1}{\beta}\)

So, \(\displaystyle {\lambda}e^{{-\lambda}x}\) becomes

\(\displaystyle \frac{1}{\beta}e^{\frac{-x}{\beta}}\)

Beta is called the scale parameter and lambda the rate parameter.

5-8 hour shifts is 40 hours. The probability it does not break down in this

time can then be found by integrating over 0 to 40 and subtracting from 1

\(\displaystyle 1-\frac{1}{50}\int_{0}^{40}e^{\frac{-x}{50}}dx\)

Or, if the machine does not break down within 40 hours, then it is sure to breakdown some time after that. Whenever that may be.

\(\displaystyle \frac{1}{50}\int_{40}^{\infty}e^{\frac{-x}{50}}dx\)
 
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May 2010
39
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if the no of machine's that fails during 5 consecutive 8 hour shifts may be modeled by a Poisson distribution

what will the value of lambda be

Please help.
 
May 2010
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In your beta notation:

\(\displaystyle \lambda = \beta^{-1}\)


Reasoning:
If you write your exponential pdf in the form
\(\displaystyle f(x)=ae^{-ax}\)

Then it models the time to failure of a poisson process with parameter \(\displaystyle a\). Your pdf is in that form with \(\displaystyle a=1/\beta\)

This is why you will often see the exponential distribution written as \(\displaystyle exp(\lambda)\). in fact i have never seen \(\displaystyle exp(\beta)\) until you asked this question.
 

matheagle

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I would think is a binomial with n=5 (the shifts)
Where p is the probability of not failing during an eight hour period, which you can calculate via the exponential distribution.
But the exponential denisty can be written either way.
So I don't know where the parameter belongs.