Estimating the parameter $\beta$ The lifetime of computer monitors has a exponential distribution where the expected value can be written as:
$\mu(s) = \frac{\beta}{s}$
Where $s$ is how bright the monitor is and both $s$ and $\beta$ are >0
How do I estimate the maximum-likelihood of parameter $\beta$?
I know that the log-likelihood for a exponential distribution is:
$l(\lambda) = n(\ln(\lambda) - \lambda\bar{t})$
And taking the derivative and equals it to zero yields:
$\frac{dl(\lambda)}{d\lambda} = n(\frac{1}{\lambda} - \bar{t}) = 0$
$\bar{t} = \frac{1}{\lambda}$
Here is where I get stuck.
EDIT 1
I have data for both the lifetime of the computer monitors and their brightness.

To estimate the $\beta$ by the maximum likelihood method, let $Y_1,\ldots, Y_n$ be the sample of lifetimes, with $Y_i\sim \text{Exp}(\beta/s_i)$, s.t. $E(Y_i) = \beta/s_i$ and independently for each $i$, where $s_i$ is the monitor brightness.
Then the likelihood function for $\beta$ is given by
$$
L(\beta) = \prod_{i=1}^n (s_i/\beta) e^{-\frac{1}{\beta}s_i Y_i} \propto \beta^{-n} e^{-\frac{1}{\beta}\sum_is_iY_i}.
$$
I leave the rest to you...
