Currently, I have a likelihood function in the following form:


with $x_i$ taking on values of $0$ or $1$, $\lambda>0$.

I have tried taking $log$ and differentiating the likelihood function, but have been unable of finding the maximum likelihood estimate of $\lambda$, as the result becomes


Can anyone give me some hints on how to do that?


1 Answer 1


Assume that the number of $x_i=0$ in your dataset is $n_0$ and the number of $x_i=1$ is $n_1$. Your likelihood function is: $$L(\lambda)=(e^{-\lambda})^{n_0}(1-e^{-\lambda})^{n_1}$$ and the log-likelihood is: $$l(\lambda)=-\lambda n_0+n_1 \ln (1-e^{-\lambda})$$ It is easy to differentiate this function to obtain the following MLE: $$\lambda = \ln \frac{n_0+n_1}{n_0}$$


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