Represent mode of a probability distribution.

Question

I am finding it difficult to represent a simple analysis done in Matlab in equation form. The operation is of two step.

1. Bin a data in 10 bins. (the distribution is unimodal) and
2. then find the bin with maximum density.

In other words finding the mode of a distribution.

Answer 1

Let $\hat{p}(x)$ be the empirical distribution of your data set, i.e. $\hat{p}(x) = \frac{ \text{ # of observed x}}{\text{# of observations}}$.

Then, the mode is $\arg\max_x \hat{p}(x)$.

For a probability density or mass function $f(x)$, the mode is $\arg \max_{x} f(x)$.