Maximum Likelihood (MLE) is a favorite interview topic at many companies and their interviewers, and in this short post, we are going to discuss two simple problems involving MLEs.

*In [previous post_](https://medium.com/swlh/linear-regression-and-maximum-likelihood-1dcb9435c71e)*, _we saw how we can derive Maximum Likelihood Estimator of a linear regression problem, and showed that it is same as the least-squared-error(LSE) estimator. Maximum Likelihood (MLE) is a favorite interview topic at many companies and their interviewers, and in this short post, we are going to discuss two simple problems involving MLEs.

- Start with the data points and distribution given in the problem statement. Always remember that the data points are fixed (constants), and the
*parameters*of the random distribution are the variables to be*estimated*in these problems. So, for example, for a normal distribution, the data points {(x₁, y₁), (x₂, y₂), …, } form the constants, and the mean/standard deviation of the distribution form the parameters to be estimated. - Write the expression for
*probability distribution _(More formally, the _probability density function (pdf)*) for a single data point. - Assuming data points to be independent, write the expression for the _joint probability distribution _by multiplying the individual probability density functions. (NOTE: We are yet to encounter an interview problem where different data points are not independent). This joint distribution is also called the _likelihood _of the data.
- Maximize the likelihood. Use _log likelihood _if it simplifies the calculation.

With this general outline, let’s now solve a few interview problems on MLE.

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