maximum likelihood estimate for lambda

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

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 that the maximum likelihood estimate for lambda is lamdba = m /(sum of ti + (n-m)*t0)?

Any help is appreciated!

Re: maximum likelihood estimate for lambda

Hey JDAWES.

The first thing you have to do is get the function and then the log-likelihood function and find the maximum using calculus.

If you have a sample of <t1,t2,...,tn> then the =likelihood will be P(T = t1)*P(T = t2)*..P(T = tn) and taking the log of this gives the log-likelihood.

Re: maximum likelihood estimate for lambda

Hey Chiro.

I have found the maximum likelihood of the function f(t) = lambda*exp^(-lambda*t) to be n/sum of t(i) which is also equal to 1/(t bar) or (1/the mean t)

Now i am stuck and i cant make the max likelihood equal what is required!