1. ## Exponential distribution.

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 ?

2. 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.

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

3. 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

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

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.

5. 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.