# Exponential distribution.

#### ilikec

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 ?

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#### galactus

MHF Hall of Honor
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.

$$\displaystyle \frac{1}{50}\int_{40}^{\infty}e^{\frac{-x}{50}}dx$$

• ilikec

#### ilikec

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

#### SpringFan25

$$\displaystyle \lambda = \beta^{-1}$$

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

MHF Hall of Honor
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.