# Maximum Likelihood estimate of $\theta = p^2$ for Bernoulli distribution

Question:

For a Bernoulli population, show that the maximum likelihood estimate of $$\theta =p^2$$ is $$\bar{x}^2$$.

I'm just looking for a hint to get started.

Obviously, I can find the MLE of a Bernoulli by maximizing the likelihood function. How can I maximize $$\theta = p^2$$ if $$p^2$$ is not in the original distribution

• Simply replace $p$ with $\sqrt{\theta}$ in the likelihood function and then maximize with respect to $\theta$. Aug 2 '20 at 17:18
• @LeanderTilstedKristensen I facepalmed. Thank you! Aug 2 '20 at 17:27
• Consider the invariance property of MLEs Aug 2 '20 at 18:06

Note that if $$\alpha$$ is the MLE of $$\beta$$ , for any function $$f(\alpha)$$ is MLE of $$f(\beta)$$. Just consider $$f(x)=x^2$$.