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I was recently asked to calculate the maximum likelihood estimate of a single Bernoulli trial. Since the MLE of a binomial distribution is just the mean of the observed number of successes, I reasoned that the MLE of a single Bernoulli trial is either $0$ or $1$, depending on whether the trial resulted in a success or not.
Is my reasoning/answer correct?
Yes, it is correct because we maximize the following data likelihood (say $x$ is the outcome of the experiment): $$f(\theta)=p(D|\theta)=p(x|\theta)=\theta^x(1-\theta)^{1-x}$$ It won’t be completely accurate if we differentiate this (and it’s not well defined when $x=0,\theta=0$ and $x=1,\theta=1$) because the function won’t be differentiable in general at the ML answer; and we don’t need to. Leaving the general definition aside, we’ll consider case by case, since $x$ is finite. If $x=1$, the likelihood becomes $f(\theta)=\theta$, and this is maximized when $\theta=1$; If $x=0$, $f(\theta)=1-\theta$, which is maximized when $\theta=0$.
