# Thread: help with exponential distribution question

1. ## help with exponential distribution question

The lifetime of a certain component may be modelled by the exponential distribution with a probability density function f(t) = lambda * e^(-lambda*t).

Show that for this distribution P(T>t) = e^(-lambda*t)?

Any help would be very appreciated!!

Thanks

2. ## Re: help with exponential distribution question

$\displaystyle f_T(t) = \lambda e^{-\lambda t}, t>0$

$\displaystyle F_T(t) = \int^{x=t}_{x=0}\lambda e^{-\lambda x}dx = \frac{\lambda e^{-\lambda x}}{-\lambda}|^{x=t}_{x=0} = -e^{-\lambda x}|^{x=t}_{x=0} = e^{-\lambda x}|^{x=0}_{x=t} = 1 - e^{-\lambda t}$

$\displaystyle P(T>t) = 1 - F_T(t) = 1 - \left(1 - e^{-\lambda t}\right) = e^{-\lambda t}$

3. ## Re: help with exponential distribution question

this follows on from the previous question;

In a test n items were used, m items failed, having times to failure t1, t2,...tm.
The remaining items were still working when the test was terminated at time t0.

Show the maximum likelihood estimate for lambda is m/ ((sum of t i) +(n-m)*t0)?

I'v been doing this question and iv got up to the point where you need to substitute in the test: the likelihood is n/sum of ti but dont know how to do the rest?