# How do I write a log likelihood function when I have 2 mean values for my pdf?

I have been given the following pdf :

fT (t; B, C) = ( exp(-t/C) - exp(-t/B) ) / ( C - B ) , (t>0) where the overall mean is B+C.

I am unsure as to how to write the log likelihood function of B and C.

The next part of the Q asks me to derive the equations that would have to be solved in order to find the max likelihood estimators of B and C.

I would be grateful for any help =)

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By definition, the log-likelihood is given by $$\ln \mathcal{L}(B,C|x_1 , \ldots ,x_n ) = \sum\limits_{i = 1}^n {\ln f(x_i|B,C)}.$$ Thus, in our example, $$\ln \mathcal{L}(B,C|x_1 , \ldots ,x_n ) = \sum\limits_{i = 1}^n {\ln \bigg[\frac{{e^{ - x_i /C} - e^{ - x_i /B} }}{{C - B}}\bigg]} .$$

EDIT: In view of the next part of the question, it may be useful to write $$\ln \mathcal{L}(B,C|x_1 , \ldots ,x_n ) = \sum\limits_{i = 1}^n {\ln [e^{ - x_i /C} - e^{ - x_i /B} ]} - n\ln (C - B).$$

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I have just realised for my Q I first need to write down the likelihood function. How would I do this? –  user5324 Jan 2 '11 at 22:02
The likelihood function is given by $\mathcal{L}(B,C|x_1 , \ldots ,x_n ) = \prod\nolimits_{i = 1}^n {f(x_i |B,C)}$. –  Shai Covo Jan 2 '11 at 22:06
Evaluate the log-likelihood and derive the equations that would have to be solved in order to find the Maximum Likelihood Estimators for B and C. This is the Q i was given, for the second part where it says to solve the estimators for B and C, would you take that as having to solve B and C seperately, or as a whole? Thanks =) –  user5324 Jan 2 '11 at 22:24
Wouldn't it be enough to write the equations corresponding to $\frac{\partial }{{\partial C}}\ln \mathcal{L}(B,C|x_1 , \ldots ,x_n ) = 0$ and $\frac{\partial }{{\partial B}}\ln \mathcal{L}(B,C|x_1 , \ldots ,x_n ) = 0$? –  Shai Covo Jan 2 '11 at 22:44